#7503828

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2146 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,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, 7411024);