#7527841

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#36020	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1998 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=i,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7435037);