#630313

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2619	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3002 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)(13,14,16,17)(15,19,20,18);(6,9)(10,11)(14,17)(18,19)(22,25)(26,28);(1,2,3,4)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,25)(27,28);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23,24,27)(22,26,25,28);(5,7,8,12)(6,10,9,11)(13,15,16,20)(14,18,17,19)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=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, 537509);