#7557335

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,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,e^b=ei,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, 7464531);