#7517028

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	C₂×#3269	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2027 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,d^c=di,e^c=ei,g^c=gi,h^c=hi,g^d=gi,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, 7424224);