#7674702

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

5018 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7581898);