#688663

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4413 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=h,c^2=h,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, 595859);