#636828

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

3330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,4)(6,8)(11,15)(14,16)(17,18,22,23)(19,24,25,20)(21,29,30,26)(27,28,32,31);(10,13)(14,16)(18,23)(19,25)(26,29)(27,32);(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,29)(30,32);(1,2)(3,4)(9,11,12,15)(10,14,13,16)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27);(17,21,22,30)(18,26,23,29)(19,27,25,32)(20,28,24,31)

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=h,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, 544024);