#692976

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	#14716	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3694 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)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18,21,22)(19,20,24,23);(6,8)(10,13)(14,15)(18,22)(20,23);(5,6)(7,8)(11,16)(14,15)(18,22)(19,24);(1,2)(3,4)(9,11,12,16)(10,14,13,15)(17,19,21,24)(18,23,22,20);(9,10)(11,14)(12,13)(15,16)(17,20,21,23)(18,24,22,19)

Polycyclic presentation:

2^9|a^2=gh,d^2=f,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, 600172);