#7718112

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

4667 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=i,f^2=g,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, 7625308);