#6467903

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	#35569	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1888 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375099);