#692062

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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	#16250	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3190 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)(6,8)(9,10)(11,12)(13,14)(15,16)(18,22)(19,24);(6,8)(9,10,13,14)(11,12,15,16);(5,6,7,8)(10,14)(11,15)(17,18,21,22)(19,23,24,20);(1,2)(3,4)(9,11)(10,12)(13,15)(14,16)(17,19)(18,23)(20,22)(21,24);(9,12)(10,11)(13,16)(14,15)(17,20,21,23)(18,19,22,24)

Polycyclic presentation:

2^9|b^2=f,c^2=fg,e^2=gh,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, 599258);