#7515814

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	#25538	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1832 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)(7,12)(8,10)(9,11)(13,14,18,22)(15,23,26,39)(16,27,30,40)(17,31,32,21)(19,33,36,24)(20,34,38,28)(25,35,29,37)(41,42,44,43);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(15,24)(16,28)(17,32)(19,23)(20,27)(21,31)(26,33)(30,34)(36,39)(38,40)(41,44)(42,43);(5,8,9,7)(6,10,11,12)(13,15)(14,19)(16,29)(17,33)(18,26)(20,37)(21,39)(22,36)(23,31)(24,32)(25,30)(27,44)(28,43)(34,42)(35,38)(40,41);(2,4)(13,16,18,30)(14,20,22,38)(15,25,26,29)(17,34,32,28)(19,35,36,37)(21,40,31,27)(23,41,39,44)(24,42,33,43);(6,11)(10,12)(15,26)(19,36)(23,39)(24,33)(25,29)(35,37)(41,44)(42,43)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=i,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,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

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7423010);