#705154

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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	C₂²×#546	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5804 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:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=f,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, 612350);