#7715913

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

4942 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7623109);