#7561603

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

2200 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)(13,14,18,22)(15,19,26,33)(16,20,27,34)(17,21,29,36)(23,30,38,42)(24,31,25,32)(28,35,40,44)(37,41,39,43);(2,3)(6,10)(11,12)(13,15,18,26)(14,19,22,33)(16,23,27,38)(17,24,29,25)(20,30,34,42)(21,31,36,32)(28,39,40,37)(35,43,44,41);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,18,27)(14,20,22,34)(15,23,26,38)(17,28,29,40)(19,30,33,42)(21,35,36,44)(24,39,25,37)(31,43,32,41);(1,3,4,2)(5,8,9,7)(6,12,10,11)(15,24)(16,27)(17,29)(19,31)(20,34)(21,36)(23,37)(25,26)(30,41)(32,33)(38,39)(42,43);(6,10)(11,12)(16,27)(20,34)(23,38)(28,40)(30,42)(35,44)(37,39)(41,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7468799);