#7630853

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4650 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538049);