#7521845

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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2045 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)(5,6,9,10)(7,11,8,12)(13,14,19,24)(15,20,16,21)(17,22,31,40)(18,23,33,42)(25,34,27,36)(26,35,28,37)(29,38,43,44)(30,39,32,41);(1,2)(3,4)(6,10)(11,12)(13,15,19,16)(14,20,24,21)(17,29,31,43)(18,28,33,26)(22,38,40,44)(23,37,42,35)(25,32,27,30)(34,41,36,39);(2,4)(13,16,19,15)(14,21,24,20)(17,27,31,25)(18,28,33,26)(22,36,40,34)(23,37,42,35)(29,30,43,32)(38,39,44,41);(2,4)(5,7,9,8)(6,11,10,12)(13,17,19,31)(14,22,24,40)(15,25,16,27)(18,32,33,30)(20,34,21,36)(23,41,42,39)(26,43,28,29)(35,44,37,38);(5,8,9,7)(6,11,10,12)(15,16)(20,21)(25,27)(26,28)(29,43)(34,36)(35,37)(38,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=fi,e^2=f,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, 7429041);