#7521397

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

1776 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)(13,14,19,24)(15,25,28,34)(16,30,33,37)(17,23,18,22)(20,26,36,29)(21,31,38,32)(27,35,41,44)(39,42,43,40);(1,2)(3,4)(6,10)(11,12)(15,26)(16,31)(17,18)(20,34)(21,37)(22,23)(25,36)(28,29)(30,38)(32,33)(39,43)(40,42);(13,15,19,28)(14,20,24,36)(16,27,33,41)(17,26,18,29)(21,35,38,44)(22,34,23,25)(30,39,37,43)(31,40,32,42);(2,4)(5,7,9,8)(6,11,10,12)(13,16,19,33)(14,21,24,38)(15,27,28,41)(17,31,18,32)(20,35,36,44)(22,37,23,30)(25,39,34,43)(26,40,29,42);(5,8,9,7)(6,11,10,12)(15,28)(20,36)(25,34)(26,29)(27,41)(35,44)(39,43)(40,42)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=fi,e^2=f,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,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, 7428593);