#7708634

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

4996 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)(9,10,13,12)(11,15,16,14)(17,18)(19,23)(20,29)(21,30)(22,24)(25,28)(26,31)(27,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,14,12,15)(17,19,22,27)(18,23,24,32)(20,21,28,31)(25,26,29,30);(2,5)(6,8)(9,10,13,12)(11,14,16,15)(18,24)(19,27)(20,28)(26,30);(9,12,13,10)(11,15,16,14)(17,18)(19,23)(20,25)(21,26)(22,24)(27,32)(28,29)(30,31);(1,2)(3,6)(4,5)(7,8)(17,20,22,28)(18,25,24,29)(19,21,27,31)(23,26,32,30);(1,3)(2,6)(4,7)(5,8)(17,21)(18,26)(19,28)(20,27)(22,31)(23,29)(24,30)(25,32)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=g,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, 7615830);