#8699069

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	#42422	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4848 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=gh,e^2=g,f^2=gi,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, 8606265);