#996130

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3938 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)(9,10,13,14)(11,12,16,15)(17,18,22,23)(19,21,20,24);(6,8)(10,14)(11,16)(18,23)(19,20);(5,6)(7,8)(10,14)(12,15)(17,19,22,20)(18,24,23,21);(1,2)(3,4)(5,6)(7,8)(9,11)(10,15)(12,14)(13,16)(17,20,22,19)(18,24,23,21);(9,12,13,15)(10,11,14,16)(17,21,22,24)(18,19,23,20)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 903326);