#6460020

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2241 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 6367216);