#7718105

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

4819 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:

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

Polycyclic presentation:

2^9|b^2=h,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, 7625301);