#6463229

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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2471 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)(7,12)(8,10)(9,11)(13,14,20,26)(15,27,31,39)(16,32,33,38)(17,23,35,42)(18,25,19,24)(21,30,40,29)(22,28,41,34)(36,44,37,43);(1,2,3,4)(7,8)(10,12)(13,15,19,30,20,31,18,29)(14,21,25,39,26,40,24,27)(16,17,34,37,33,35,28,36)(22,23,32,44,41,42,38,43);(5,7,9,8)(6,10,11,12)(13,16,19,34,20,33,18,28)(14,22,25,32,26,41,24,38)(15,17,30,37,31,35,29,36)(21,23,39,44,40,42,27,43);(5,8,9,7)(6,10,11,12)(13,17)(14,23)(15,28)(16,30)(18,36)(19,37)(20,35)(21,38)(22,39)(24,43)(25,44)(26,42)(27,41)(29,33)(31,34)(32,40);(16,33)(17,35)(22,41)(23,42)(28,34)(32,38)(36,37)(43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=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, 6370425);