#774163

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 681359);