#7557305

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1755 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei,e^d=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, 7464501);