#6457674

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2089 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6)(7,11)(8,12)(9,10)(13,14,18,23,19,24,32,34)(15,25,17,22,27,39,31,20)(16,21,30,38,29,37,40,43)(26,35,28,36,41,44,42,33);(6,10)(11,12)(13,15,19,27)(14,20,24,22)(16,28,29,42)(17,18,31,32)(21,35,37,44)(23,39,34,25)(26,40,41,30)(33,43,36,38);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,16,18,30,19,29,32,40)(14,21,23,38,24,37,34,43)(15,26,17,28,27,41,31,42)(20,33,25,35,22,36,39,44);(5,8,9,7)(6,12,10,11)(13,17,18,27,19,31,32,15)(14,22,23,39,24,20,34,25)(16,28,30,41,29,42,40,26)(21,36,38,44,37,33,43,35);(16,29)(21,37)(26,41)(28,42)(30,40)(33,36)(35,44)(38,43)

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,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6364870);