#6627346

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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

1866 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)(13,14,19,22)(15,23,26,27)(16,20,18,25)(17,21,28,24);(6,8)(10,11)(13,15,16,21,19,26,18,24)(14,17,25,27,22,28,20,23);(2,4)(5,6,9,8)(7,11,12,10)(13,16,19,18)(14,20,22,25)(15,24,26,21)(17,27,28,23);(5,7,9,12)(6,10,8,11)(13,17,19,28)(14,21,22,24)(15,25,26,20)(16,27,18,23);(5,8,9,6)(7,10,12,11)(15,26)(17,28)(21,24)(23,27)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=h,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6534542);