#648001

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
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

2980 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)(9,10,14,13)(11,16,12,15)(18,22)(19,24);(6,8)(9,11,14,12)(10,15,13,16);(5,6)(7,8)(9,11,14,12)(10,16,13,15)(17,18)(19,23)(20,24)(21,22);(1,2)(3,4)(9,12,14,11)(10,15,13,16)(17,19,21,24)(18,23,22,20);(9,13,14,10)(11,15,12,16)(17,20,21,23)(18,24,22,19)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,d^2=fg,e^2=fg,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, 555197);