#656957

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2145 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=f,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

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, 564153);