#6617845

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1812 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)(13,14,19,16)(15,22,25,28)(17,21,18,26)(20,27,23,24);(6,10)(8,11)(13,15,17,27,19,25,18,24)(14,20,26,28,16,23,21,22);(2,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,23)(16,19)(17,26)(18,21)(20,25)(22,27)(24,28);(5,7,9,12)(6,11,10,8)(13,16,19,14)(15,23,25,20)(17,21,18,26)(22,24,28,27);(5,8,9,11)(6,7,10,12)(13,17,19,18)(14,21,16,26)(15,24,25,27)(20,28,23,22)

Polycyclic presentation:

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

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, 6525041);