#687822

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3762 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)(6,7)(13,14,18,22)(15,16,20,19)(17,21,26,28)(23,24,25,27);(10,12)(13,15,18,20)(14,19,22,16)(17,25,26,23)(21,24,28,27);(5,6,8,7)(9,10,11,12)(13,15)(14,19)(16,22)(17,23)(18,20)(21,27)(24,28)(25,26);(1,2)(3,4)(5,7,8,6)(13,16,18,19)(14,20,22,15)(17,24,26,27)(21,25,28,23);(13,17,18,26)(14,21,22,28)(15,23,20,25)(16,24,19,27)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=gh,d^2=g,e^2=f,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, 595018);