#7562677

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	#36060	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2259 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=gi,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, 7469873);