#7527546

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	#45092	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2005 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,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^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, 7434742);