#6384032

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#34681	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2807 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)(5,6)(7,8)(9,10,12,16,13,14,20,21)(11,17,19,24,18,15,22,23);(1,2)(3,4)(11,18)(15,17)(19,22)(23,24);(6,8)(11,18)(15,17)(19,22)(23,24);(2,4)(10,14)(15,17)(16,21)(23,24);(9,11,12,19,13,18,20,22)(10,15,16,23,14,17,21,24)

Polycyclic presentation:

2^9|a^2=h,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,d^b=df,e^b=ei,e^c=ei

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, 6291228);