#7688510

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

10749 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)(11,19,21,16)(13,17,20,22)(14,18,24,23);(5,6)(7,8)(9,10,15,12)(11,19,21,16)(13,17,20,22)(14,18,24,23);(2,3)(9,11,15,21)(10,16,12,19)(13,14,20,24)(17,23,22,18);(1,3,4,2)(9,12,15,10)(11,16,21,19)(13,22,20,17)(14,23,24,18);(9,13)(10,17)(11,14)(12,22)(15,20)(16,23)(18,19)(21,24);(9,14)(10,18)(11,20)(12,23)(13,21)(15,24)(16,17)(19,22)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,d^2=h,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, 7595706);