#7779663

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

5516 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)(18,21)(22,24);(1,2,5,4)(3,6,7,8)(9,11,13,12)(10,14,16,15)(17,18,20,21)(19,22,23,24);(2,4)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,20,23)(18,22,21,24);(3,7)(6,8)(9,10,13,16)(11,14,12,15);(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=h,b^2=gi,c^2=g,d^2=h,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	#72304381, #72304384, #72304437
maximal normal subgroup of	#72304381, #72304384, #72304437

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7686859);