#6457668

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2068 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fg,d^2=f,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,d^c=di

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, 6364864);