#6627429

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	#24762	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1877 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)(5,6,9,8)(7,10,12,11)(13,14,19,23)(15,17,22,18)(16,21,27,20)(24,26,25,28);(6,8)(10,11)(13,15,18,14,19,22,17,23)(16,25,26,21,27,24,28,20);(2,4)(5,6,9,8)(7,11,12,10)(13,16,19,27)(14,20,23,21)(15,24,22,25)(17,28,18,26);(5,7,9,12)(6,10,8,11)(13,16)(14,21)(15,25)(17,28)(18,26)(19,27)(20,23)(22,24);(5,8,9,6)(7,10,12,11)(13,17,19,18)(14,22,23,15)(16,26,27,28)(20,24,21,25)

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=h,d^2=hi,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, 6534625);