#7682048

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

5043 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)(9,10,13,15)(11,12,14,16)(17,18)(19,25)(20,24)(21,23)(22,26)(27,32)(28,31)(29,30);(1,3,5,8)(2,6,4,7)(9,11,13,14)(10,12,15,16)(17,19)(18,22)(20,28)(21,26)(23,25)(24,32)(27,30)(29,31);(9,10)(11,12)(13,15)(14,16)(18,23)(19,26)(24,29)(27,28);(9,12)(10,14)(11,15)(13,16)(17,18)(19,22)(20,29)(21,23)(24,30)(25,26)(27,32)(28,31);(1,4,5,2)(3,7,8,6)(17,20,21,30)(18,24,23,29)(19,27,26,28)(22,31,25,32);(17,19)(18,22)(20,27)(21,26)(23,25)(24,31)(28,30)(29,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,e^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=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, 7589244);