#649780

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4588 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)(18,20)(19,24);(10,13)(14,15)(17,18,22,20)(19,21,24,23);(5,6)(7,8)(17,19,22,24)(18,23,20,21);(1,2)(3,4)(5,6)(7,8)(9,11,12,16)(10,14,13,15)(17,20,22,18)(19,21,24,23);(9,10)(11,14)(12,13)(15,16)(17,21)(18,19)(20,24)(22,23)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 556976);