#7687866

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

11003 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)(5,6,7,8)(9,10,15,12)(11,13,16,18)(14,17,23,20)(19,21,24,22);(5,6)(7,8)(9,11)(10,13)(12,18)(14,19)(15,16)(17,21)(20,22)(23,24);(2,3)(10,12)(11,16)(17,20)(19,24);(1,3,4,2)(9,12,15,10)(11,18,16,13)(14,20,23,17)(19,22,24,21);(9,13,15,18)(10,16,12,11)(14,22,23,21)(17,19,20,24);(9,14,15,23)(10,17,12,20)(11,19,16,24)(13,21,18,22)

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7595062);