#8828866

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

5567 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303495, #72305957
maximal normal subgroup of	#72303495, #72305957
maximal quotient of	#72303506, #72305964

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8736062);