#7501805

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2367 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7409001);