#362117

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1334 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269313);