#759374

order	512
nilpotency class	2
isoclinism	isoclinic class of #759288
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2822 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)(9,10,14,16)(11,13,12,15)(17,18,22,20)(19,24,21,23);(6,8)(9,10)(11,13)(12,15)(14,16)(17,19,22,21)(18,23,20,24);(5,6)(7,8)(9,11,14,12)(10,15,16,13)(17,20,22,18)(19,23,21,24);(1,2)(3,4)(9,12,14,11)(10,15,16,13);(9,13,14,15)(10,11,16,12)(17,21,22,19)(18,24,20,23)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=f,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,e^c=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, 666570);