#7503167

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	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2702 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)(8,10)(9,12)(13,14)(15,20)(16,21)(17,31)(18,22)(19,23)(24,33)(25,40)(26,34)(27,35)(28,36)(29,38)(30,39)(32,37)(41,44)(42,43);(5,7)(6,9)(8,12)(10,11)(13,15,19,27)(14,20,23,35)(16,28,30,42)(17,26,18,25)(21,36,39,43)(22,40,31,34)(24,32,41,29)(33,37,44,38);(2,4)(13,16,19,30)(14,21,23,39)(15,24,27,41)(17,29,18,32)(20,33,35,44)(22,37,31,38)(25,28,26,42)(34,43,40,36);(2,4)(6,10)(7,12)(13,14)(15,20)(16,21)(17,31)(18,22)(19,23)(24,33)(25,40)(26,34)(27,35)(28,36)(29,38)(30,39)(32,37)(41,44)(42,43);(13,17,19,18)(14,22,23,31)(15,25,27,26)(16,29,30,32)(20,34,35,40)(21,37,39,38)(24,28,41,42)(33,43,44,36)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei

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, 7410363);