#126533

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₄×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₄²
commutator	C₄	C₄²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1726
prime radical	C₂³	#405
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

316 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(9,10,13,18,14,19,29,40,15,20,30,41,31,42,53,64)(11,16,22,33,24,35,46,57,23,34,45,56,47,58,66,70)(12,17,26,37,28,39,51,62,27,38,50,61,52,63,68,72)(21,32,43,54,25,36,48,59,44,55,65,69,49,60,67,71);(1,2,3,5,4,6,7,8)(9,11)(10,16)(12,25,27,49)(13,22)(14,24)(15,23)(17,36,38,60)(18,33)(19,35)(20,34)(21,28,44,52)(26,48,50,67)(29,46)(30,45)(31,47)(32,39,55,63)(37,59,61,71)(40,57)(41,56)(42,58)(43,51,65,68)(53,66)(54,62,69,72)(64,70);(9,12)(10,17)(11,21,23,44)(13,26)(14,28)(15,27)(16,32,34,55)(18,37)(19,39)(20,38)(22,43,45,65)(24,25,47,49)(29,51)(30,50)(31,52)(33,54,56,69)(35,36,58,60)(40,62)(41,61)(42,63)(46,48,66,67)(53,68)(57,59,70,71)(64,72)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,g^2=i,c^b=cg,f^b=fi,e^c=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210650, #72210687, #72234252
maximal normal subgroup of	#72210650, #72210687, #72234252
maximal quotient of	#72210602, #72210619

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33729);