#7522476

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	#45248	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2148 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)(15,21)(16,20)(17,22)(18,23)(19,24)(25,36)(26,37)(27,34)(28,35)(29,44)(30,39)(31,40)(32,41)(33,42)(38,43);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,20)(16,18)(17,29)(19,28)(21,23)(22,38)(24,37)(25,31)(26,32)(27,33)(30,43)(34,40)(35,41)(36,42)(39,44);(2,4)(13,16,18,15)(14,21,23,20)(17,27,30,25)(19,28,32,26)(22,36,39,34)(24,37,41,35)(29,31,43,33)(38,40,44,42);(2,4)(5,7,9,8)(6,11,12,10)(13,17,18,30)(14,22,23,39)(15,25,16,27)(19,33,32,31)(20,34,21,36)(24,42,41,40)(26,43,28,29)(35,44,37,38);(5,8,9,7)(6,11,12,10)(14,23)(15,16)(22,39)(24,41)(25,27)(26,28)(29,43)(40,42)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=f,e^2=fi,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, 7429672);