#7522421

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	#45241	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2180 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)(8,10)(9,12)(13,14,17,21)(15,23,25,19)(16,27,29,20)(18,22,30,35)(24,32,40,37)(26,38,41,33)(28,36,31,34)(39,43,42,44);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(16,28)(18,30)(20,34)(22,35)(24,39)(26,41)(27,36)(29,31)(32,43)(33,38)(37,44)(40,42);(2,4)(13,15,17,25)(14,19,21,23)(16,24,29,40)(18,26,30,41)(20,32,27,37)(22,33,35,38)(28,39,31,42)(34,43,36,44);(2,4)(5,7,9,8)(6,11,12,10)(13,16,17,29)(14,20,21,27)(15,24,25,40)(18,31,30,28)(19,32,23,37)(22,36,35,34)(26,42,41,39)(33,44,38,43);(5,8,9,7)(6,11,12,10)(15,25)(19,23)(24,40)(26,41)(32,37)(33,38)(39,42)(43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=fi,e^2=f,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, 7429617);