#8696210

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	#43029	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4685 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)(3,6)(4,7)(5,8)(9,10,13,11)(12,15,16,14)(17,18,24,28)(19,26,20,25)(21,23,22,27)(29,32,30,31);(1,3)(2,5)(4,8)(6,7)(10,11)(14,15)(17,19,24,20)(18,25,28,26)(21,29,22,30)(23,31,27,32);(9,11,13,10)(12,15,16,14)(17,20,24,19)(18,26,28,25)(21,30,22,29)(23,32,27,31);(1,2,4,7)(3,6,8,5)(9,12,13,16)(10,14,11,15)(17,21,24,22)(18,27,28,23)(19,29,20,30)(25,32,26,31);(1,3)(2,6)(4,8)(5,7)(17,22,24,21)(18,23,28,27)(19,30,20,29)(25,31,26,32);(1,2)(3,5)(4,7)(6,8)(17,23)(18,22)(19,31)(20,32)(21,28)(24,27)(25,30)(26,29)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8603406);