#917588

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2388	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3126 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)(7,11)(8,9)(10,12)(14,18)(16,19)(21,22,25,26)(23,28,27,24);(6,9)(10,11)(13,14,17,18)(15,19,20,16)(21,23)(22,24)(25,27)(26,28);(6,9)(10,11)(14,18)(15,20)(22,26)(23,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,18)(17,20)(21,22)(23,28)(24,27)(25,26);(5,7,8,12)(6,10,9,11)(13,16,17,19)(14,15,18,20)(21,24,25,28)(22,27,26,23)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 824784);