#7521468

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

1787 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)(7,11)(8,12)(9,10)(13,14,18,23)(15,25,28,20)(16,30,27,44)(17,33,26,40)(19,36,35,24)(21,34,41,32)(22,38,42,31)(29,39,37,43);(1,2)(3,4)(6,10)(11,12)(13,15,18,28)(14,20,23,25)(16,31,27,38)(17,34,26,32)(19,37,35,29)(21,40,41,33)(22,30,42,44)(24,43,36,39);(13,16,18,27)(14,21,23,41)(15,26,28,17)(19,34,35,32)(20,22,25,42)(24,44,36,30)(29,31,37,38)(33,39,40,43);(2,4)(5,7,9,8)(6,11,10,12)(13,17,18,26)(14,22,23,42)(15,27,28,16)(19,38,35,31)(20,21,25,41)(24,40,36,33)(29,32,37,34)(30,39,44,43);(5,8,9,7)(6,11,10,12)(14,23)(15,28)(16,27)(22,42)(24,36)(29,37)(32,34)(33,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,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, 7428664);