#758754

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

3008 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)(5,6,9,8)(7,11,12,10)(14,15)(18,19)(21,22)(23,25)(24,26)(27,28);(13,14,17,15)(16,19,20,18)(21,22,24,26)(23,25,27,28);(1,2)(3,4)(14,15)(18,19)(23,27)(25,28);(5,7,9,12)(6,10,8,11)(13,15,17,14)(16,19,20,18)(21,23)(22,25)(24,27)(26,28);(5,8,9,6)(7,11,12,10)(13,16)(14,18)(15,19)(17,20)(21,22)(23,25)(24,26)(27,28)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,d^2=i,e^2=hi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

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, 665950);