#6483489

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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×#3176	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2082 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)(15,27)(16,32)(17,23)(18,24)(19,25)(20,26)(21,29)(22,33)(28,41)(30,40)(31,39)(34,38)(35,42)(36,43)(37,44);(1,2,3,4)(7,8)(10,12)(13,15,18,29,20,31,19,30)(14,21,25,27,26,40,24,39)(16,17,28,36,34,37,33,35)(22,23,38,42,41,44,32,43);(5,7,9,8)(6,10,11,12)(13,16,18,28,20,34,19,33)(14,22,25,38,26,41,24,32)(15,17,29,36,31,37,30,35)(21,23,27,42,40,44,39,43);(5,8,9,7)(6,10,11,12)(13,17)(14,23)(15,28)(16,30)(18,36)(19,35)(20,37)(21,38)(22,39)(24,43)(25,42)(26,44)(27,41)(29,34)(31,33)(32,40);(6,11)(10,12)(13,18,20,19)(14,24,26,25)(15,29,31,30)(16,33,34,28)(17,35,37,36)(21,39,40,27)(22,38,41,32)(23,42,44,43)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,d^c=di,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, 6390685);