#7523078

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1824 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7430274);