#9040967

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

6466 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72340989, #72340995, #72341004
maximal subgroup of	#72265473, #72265778, #72333431, …
maximal normal subgroup of	#72265473, #72265778, #72333431, …
maximal quotient of	#72266567, #72333443, #72333471, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8948163);