#7681115

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

10704 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,d^2=h,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7588311);