#688966

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

2274 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)(7,8)(9,11)(10,12)(14,16)(19,20)(21,22,26,25)(23,28,27,24);(5,7)(6,10)(8,11)(9,12)(13,14,18,16)(15,19,17,20);(6,11)(7,12)(13,15,18,17)(14,19,16,20)(21,23)(22,24)(25,28)(26,27);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,16,18,14)(15,19,17,20)(21,24,26,28)(22,27,25,23);(5,8)(6,7)(9,10)(11,12)(13,17,18,15)(14,19,16,20)(21,25,26,22)(23,28,27,24)

Polycyclic presentation:

2^9|a^2=gh,b^2=fh,c^2=fh,d^2=fg,e^2=fg,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, 596162);