#7563617

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1682 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=g,d^2=gi,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=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7470813);