#7708622

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
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4852 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7615818);