#7687138

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

4769 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7594334);