#7708773

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

5680 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)(10,14)(12,16)(17,18,22,26)(19,21,25,23)(20,29,31,24)(27,30,28,32);(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,12,15,16)(17,19,22,25)(18,23,26,21)(20,28,31,27)(24,32,29,30);(2,5)(6,8)(10,14)(11,15)(17,20,22,31)(18,24,26,29)(19,27,25,28)(21,32,23,30);(17,21,22,23)(18,25,26,19)(20,30,31,32)(24,27,29,28);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,12,14,16)(17,20)(18,24)(19,28)(21,30)(22,31)(23,32)(25,27)(26,29);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,14)(13,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,32)(29,31)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=h,e^2=gh,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, 7615969);