#149496

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₂×C₈² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₄²
commutator	C₄	C₂×C₈²
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂²	C₂×#346
prime radical	C₂³	#346
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,h^2=i,b^a=bh,e^a=ei,c^b=ci,d^b=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231854, #72231860, #72231861, …
maximal normal subgroup of	#72231854, #72231860, #72231861, …
maximal quotient of	#72234239, #72234249

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56692);