#6456476

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
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	#34977	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2121 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)(13,14,18,16,20,22,19,17)(15,23,28,21,26,27,25,24);(1,2)(3,4)(13,15,20,26)(14,21,22,24)(16,23,17,27)(18,25,19,28);(2,4)(6,9)(10,12)(13,16,18,22,20,17,19,14)(15,21,28,27,26,24,25,23);(2,4)(13,17,19,22,20,16,18,14)(15,24,25,27,26,21,28,23);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,16,22,17)(15,25,26,28)(21,23,24,27)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fi,d^2=f,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di

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, 6363672);