#7779667

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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

5800 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304364, #72304423, #72304437
maximal normal subgroup of	#72304364, #72304423, #72304437

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7686863);