#6456485

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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	#35189	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1942 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)(13,14,19,23,20,24,18,26)(15,25,27,16,28,17,22,21);(1,2)(3,4)(13,15,20,28)(14,21,24,16)(17,26,25,23)(18,27,19,22);(2,4)(6,9)(10,12)(13,16,18,25,20,21,19,17)(14,15,26,22,24,28,23,27);(2,4)(13,17,19,21,20,25,18,16)(14,22,23,15,24,27,26,28);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,23,24,26)(15,27,28,22)(16,25,21,17)

Polycyclic presentation:

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

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, 6363681);