#7514799

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	#36085	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2667 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)(7,11)(8,10)(9,12)(13,14,19,23)(15,24,27,36)(16,28,32,40)(17,22,18,33)(20,34,37,25)(21,31,39,29)(26,35,30,38)(41,42,44,43);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(15,25)(16,29)(17,18)(20,24)(21,28)(22,33)(27,34)(31,32)(36,37)(39,40)(41,44)(42,43);(13,15)(14,20)(16,30)(17,25)(18,34)(19,27)(21,38)(22,36)(23,37)(24,33)(26,32)(28,44)(29,43)(31,42)(35,39)(40,41);(2,4)(13,16,19,32)(14,21,23,39)(15,26,27,30)(17,29,18,31)(20,35,37,38)(22,40,33,28)(24,41,36,44)(25,42,34,43);(6,10)(9,11)(13,17,19,18)(14,22,23,33)(15,25,27,34)(16,31,32,29)(20,36,37,24)(21,28,39,40)(26,43,30,42)(35,41,38,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=i,e^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,e^b=ei,g^b=gi,h^b=hi,d^c=di,g^c=gi,h^c=hi,e^d=ei,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, 7421995);