#7631995

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

5114 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)(3,6,4,7)(10,13)(14,16)(17,18,24,28)(19,29,31,25)(20,22,21,26)(23,27,32,30);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,27)(21,30)(22,23)(24,31)(26,32)(28,29);(1,2)(3,6)(4,7)(5,8)(17,20,24,21)(18,26,28,22)(19,30,31,27)(23,29,32,25);(1,3,5,4)(2,6,8,7)(17,21,24,20)(18,26,28,22)(19,30,31,27)(23,25,32,29);(1,4,5,3)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,22,24,26)(18,21,28,20)(19,23,31,32)(25,30,29,27);(17,23)(18,27)(19,26)(20,29)(21,25)(22,31)(24,32)(28,30)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=h,e^2=g,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539191);