#416624

order	512
nilpotency class	3
isoclinism	isoclinic class of #394257
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35274	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323820);