#8182921

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

5822 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)(3,5)(4,6)(7,8)(17,18)(19,21)(20,23)(22,30)(24,26)(25,32)(27,28)(29,31);(2,3)(6,7)(9,10,13,14)(11,15,16,12)(17,19,24,23)(18,21,26,20)(22,27,32,31)(25,29,30,28);(4,8)(6,7)(10,14)(12,15)(17,20,24,21)(18,19,26,23)(22,28,32,29)(25,27,30,31);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,16)(14,15)(17,21,24,20)(18,19,26,23)(22,29,32,28)(25,27,30,31);(17,22,24,32)(18,25,26,30)(19,27,23,31)(20,28,21,29);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,15)(14,16)(17,23,24,19)(18,20,26,21)(22,31,32,27)(25,28,30,29)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312311, #72342319, #72342327
maximal normal subgroup of	#72312311, #72342319, #72342327

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8090117);