#6466207

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2212 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=ghi,d^2=h,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,e^d=ei

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, 6373403);