#876606

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

2392 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 783802);