#476079

order	512
nilpotency class	3
isoclinism	isoclinic class of #459487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2804
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

796 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 796 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6,11,14,10,15,9,24)(7,16,25,27,18,32,19,12)(8,13,22,28,23,30,21,29)(17,31,33,36,20,35,34,26);(5,7,10,18)(6,12)(8,20,23,17)(9,19,11,25)(13,26)(14,16)(15,27)(21,33,22,34)(24,32)(28,31)(29,35)(30,36);(1,2,3,4)(5,8)(6,13)(7,17)(9,21)(10,23)(11,22)(12,26)(14,28)(15,30)(16,31)(18,20)(19,34)(24,29)(25,33)(27,36)(32,35);(7,18)(8,21,23,22)(12,27)(13,28,30,29)(16,32)(17,33,20,34)(19,25)(26,35,36,31)

Polycyclic presentation:

2^9|a^2=egh,b^2=g,c^2=f,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=di,d^c=de,e^c=ei,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214507, #72216224, #72279655, …
maximal normal subgroup of	#72214507, #72216224, #72279655, …
maximal quotient of	#72214172

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383275);