#7520217

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1783 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)(8,12)(9,10)(13,14,20,26)(15,21,28,39)(16,31,33,43)(17,36,27,44)(18,25,19,24)(22,35,42,34)(23,37,38,32)(29,41,30,40);(1,2)(3,4)(6,10)(11,12)(13,15,20,28)(14,21,26,39)(16,32,33,37)(17,35,27,34)(18,30,19,29)(22,36,42,44)(23,43,38,31)(24,41,25,40);(5,7,9,8)(6,11,10,12)(13,16)(14,22)(15,27)(17,28)(18,35)(19,34)(20,33)(21,38)(23,39)(24,43)(25,31)(26,42)(29,32)(30,37)(36,40)(41,44);(2,4)(13,17,20,27)(14,23,26,38)(15,16,28,33)(18,37,19,32)(21,22,39,42)(24,44,25,36)(29,35,30,34)(31,40,43,41);(5,8,9,7)(6,11,10,12)(15,28)(16,33)(21,39)(22,42)(29,30)(31,43)(34,35)(40,41)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=i,e^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=cgi,d^b=dh,e^b=ei,g^b=gi,h^b=hi,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, 7427413);