#419815

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2404
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1688 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285151, #72285600, #72285668, …
maximal normal subgroup of	#72285151, #72285600, #72285668, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327011);