#7690892

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

5327 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)(3,7)(4,5)(6,8)(9,10,13,15)(11,16,12,14)(17,18)(19,28)(20,25)(21,26)(22,27)(23,24)(29,31)(30,32);(1,3,6,4)(2,7,8,5)(9,11,13,12)(10,14,15,16)(17,19,23,27)(18,22,24,28)(20,21,29,32)(25,30,31,26);(1,4,6,3)(2,7,8,5)(9,10)(11,14)(12,16)(13,15)(18,24)(19,27)(20,29)(26,30);(1,5,6,7)(2,4,8,3)(9,12,13,11)(10,14,15,16)(17,20,23,29)(18,25,24,31)(19,21,27,32)(22,30,28,26);(17,21)(18,26)(19,29)(20,27)(22,25)(23,32)(24,30)(28,31);(17,22)(18,27)(19,24)(20,30)(21,31)(23,28)(25,32)(26,29)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=h,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, 7598088);