#612844

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
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	#15123	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3680 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)(9,10,12,14)(11,15,16,13)(17,18,23,21)(19,24,22,20);(5,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,19,23,22)(18,20,21,24);(5,6)(7,8)(17,20)(18,22)(19,21)(23,24);(1,2)(3,4)(9,10)(11,13)(12,14)(15,16)(17,21,23,18)(19,24,22,20);(9,11,12,16)(10,13,14,15)(17,22,23,19)(18,24,21,20)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 520040);