#7565375

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1795 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=fg,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, 7472571);