#8445976

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

4951 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8353172);