#7551461

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	#45332	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2065 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)(7,11)(8,9)(10,12)(13,14,17,21)(15,19,25,34)(16,27,29,20)(18,22,31,38)(23,36,28,32)(24,33,26,35)(30,41,42,37)(39,44,40,43);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,28)(17,25)(18,24)(20,36)(21,34)(22,33)(23,29)(26,31)(27,32)(30,40)(35,38)(37,44)(39,42)(41,43);(6,9)(10,11)(13,16)(14,20)(15,23)(17,29)(18,30)(19,32)(21,27)(22,37)(24,39)(25,28)(26,40)(31,42)(33,43)(34,36)(35,44)(38,41);(5,7,8,12)(6,10,9,11)(15,24)(18,31)(19,33)(22,38)(23,39)(25,26)(28,40)(30,42)(32,43)(34,35)(36,44)(37,41);(1,3,4,2)(15,25)(16,29)(19,34)(20,27)(24,26)(30,42)(33,35)(37,41)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=gi,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, 7458657);