#612698

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3194 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,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, 519894);