#7561376

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

2196 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)(8,12)(9,10)(13,14,19,24)(15,20,27,34)(16,21,30,37)(17,22,18,23)(25,32,41,44)(26,33,31,38)(28,35,29,36)(39,42,40,43);(2,3)(6,10)(11,12)(13,15,19,27)(14,20,24,34)(16,25,30,41)(17,26,18,31)(21,32,37,44)(22,33,23,38)(28,40,29,39)(35,43,36,42);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16)(14,21)(15,25)(17,28)(18,29)(19,30)(20,32)(22,35)(23,36)(24,37)(26,40)(27,41)(31,39)(33,43)(34,44)(38,42);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,17)(14,22)(16,28)(18,19)(21,35)(23,24)(26,31)(29,30)(33,38)(36,37)(39,40)(42,43);(6,10)(11,12)(13,17,19,18)(14,22,24,23)(15,26,27,31)(16,29,30,28)(20,33,34,38)(21,36,37,35)(25,39,41,40)(32,42,44,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=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, 7468572);