#7682095

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	C₂×#534	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4831 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7589291);