#6384269

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#34829	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2227 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2)(3,4)(6,9)(10,12)(13,14,17,16,18,20,26,25)(15,19,23,24,21,27,28,22);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,24)(17,23)(18,21)(20,27)(22,25)(26,28);(2,4)(5,7,8,11)(6,10,9,12)(15,21)(19,27)(22,24)(23,28);(6,9)(10,12);(13,16,17,20,18,25,26,14)(15,22,23,19,21,24,28,27)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6291465);