#455809

order	512
nilpotency class	3
isoclinism	isoclinic class of #453981
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#15123	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363005);