#7692332

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

11081 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^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.

Relationships

maximal subgroup of	#72284809, #72304537, #72304562
maximal normal subgroup of	#72284809, #72304537, #72304562

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7599528);