#636018

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
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	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3754 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)(9,10,11,12)(13,14)(15,22)(16,20)(17,25)(18,19)(21,26)(23,24)(27,28);(6,8)(9,10,11,12)(14,19)(15,20)(21,25)(23,27);(5,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,23)(18,20)(19,22)(21,24)(25,28)(26,27);(1,2)(3,4)(13,16)(14,20)(15,19)(17,24)(18,22)(21,27)(23,25)(26,28);(13,17,18,26)(14,21,19,25)(15,23,20,27)(16,24,22,28)

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,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, 543214);