#7739386

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
prime radical	#35272	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5344 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)(11,12)(13,14)(15,16)(18,21)(22,24);(1,2,5,3)(4,7,8,6)(17,18,20,21)(19,22,23,24);(2,3)(6,7)(10,14)(11,15)(17,19,20,23)(18,22,21,24);(1,2,5,3)(4,6,8,7)(9,11,13,15)(10,12,14,16);(1,3,5,2)(4,7,8,6)(9,12)(10,15)(11,14)(13,16);(1,4,5,8)(2,6,3,7)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7646582);