#1294424

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2693 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1201620);