#8661380

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

5214 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2,4,6)(3,7,8,5)(9,10)(11,12)(13,14)(15,16)(18,23)(19,27)(20,26)(29,30);(1,3,4,8)(2,5,6,7)(10,14)(11,15)(17,18,22,23)(19,24,27,25)(20,21,26,31)(28,29,32,30);(1,3)(2,5)(4,8)(6,7)(9,11,13,15)(10,12,14,16)(17,19)(18,24)(20,28)(21,29)(22,27)(23,25)(26,32)(30,31);(1,3)(2,5)(4,8)(6,7)(9,12)(10,15)(11,14)(13,16)(17,19)(18,25)(20,28)(21,30)(22,27)(23,24)(26,32)(29,31);(1,2)(3,5)(4,6)(7,8)(17,20)(18,21)(19,28)(22,26)(23,31)(24,29)(25,30)(27,32);(17,21)(18,26)(19,29)(20,23)(22,31)(24,32)(25,28)(27,30)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265962, #72265963, #72265966
maximal normal subgroup of	#72265962, #72265963, #72265966
maximal quotient of	#72266541, #72266560

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8568576);