#1083604

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3232 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:

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

Polycyclic presentation:

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

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, 990800);