#920681

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(6,9)(7,11)(15,19)(18,20)(21,22,27,24)(23,29,25,28)(26,32,34,30)(31,35,33,36);(7,11)(10,12)(14,17)(18,20)(21,23,27,25)(22,28,24,29)(26,31,34,33)(30,35,32,36);(5,6)(7,10)(8,9)(11,12)(13,14,16,17)(15,18,19,20)(21,24,27,22)(23,28,25,29)(26,32,34,30)(31,35,33,36);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,15,16,19)(14,18,17,20)(21,25,27,23)(22,29,24,28)(26,33,34,31)(30,36,32,35);(21,26,27,34)(22,30,24,32)(23,31,25,33)(28,35,29,36)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=h,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 827877);