#6444978

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	#2416	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352174);