#6554712

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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	#39118	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1811 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6461908);