#6456648

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35520	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1914 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)(13,14,19,16,18,22,17,21)(15,23,27,28,26,20,24,25);(1,2)(3,4)(14,16)(15,24)(17,19)(21,22)(25,28)(26,27);(2,4)(6,9)(10,12)(13,15,17,24,18,26,19,27)(14,20,21,28,22,23,16,25);(2,4)(13,16,19,22,18,21,17,14)(15,25,27,23,26,28,24,20);(5,7,8,11)(6,10,9,12)(15,26)(20,23)(24,27)(25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6363844);