#8505711

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

4962 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)(4,7)(6,8)(9,10,12,14)(11,13,15,16)(17,18,23,28)(19,25,20,24)(21,30,31,26)(22,27,32,29);(2,3)(6,7)(9,11,12,15)(10,13,14,16)(17,19,23,20)(18,24,28,25)(21,27,31,29)(22,26,32,30);(1,2,5,3)(4,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,19,23,20)(18,25,28,24)(21,29,31,27)(22,26,32,30);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,15)(14,16)(17,20,23,19)(18,25,28,24)(21,27,31,29)(22,26,32,30);(1,4,5,8)(2,6,3,7)(9,10)(11,13)(12,14)(15,16)(17,21)(18,26)(19,29)(20,27)(22,24)(23,31)(25,32)(28,30);(17,22,23,32)(18,27,28,29)(19,26,20,30)(21,24,31,25)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412907);