#7519946

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1810 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7427142);