#7523045

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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1778 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)(5,6,9,12)(7,11,8,10)(13,14)(15,25)(16,21)(17,22)(18,33)(19,24)(20,23)(26,42)(27,36)(28,35)(29,37)(30,44)(31,41)(32,40)(34,39)(38,43);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,21)(16,19)(17,30)(18,27)(20,29)(22,38)(23,36)(24,25)(26,31)(28,34)(32,43)(33,37)(35,41)(39,42)(40,44);(2,4)(13,16,19,15)(14,21,24,25)(17,28,32,26)(18,27,20,29)(22,35,40,42)(23,37,33,36)(30,31,43,34)(38,39,44,41);(2,4)(5,7,9,8)(6,11,12,10)(13,17)(14,22)(15,26)(16,28)(18,31)(19,32)(20,34)(21,35)(23,39)(24,40)(25,42)(27,43)(29,30)(33,41)(36,38)(37,44);(5,8,9,7)(6,11,12,10)(13,18,19,20)(14,23,24,33)(15,27,16,29)(17,31,32,34)(21,36,25,37)(22,39,40,41)(26,43,28,30)(35,38,42,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,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=cgi,d^b=dh,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, 7430241);