#7520153

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1799 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)(13,14,19,25)(15,27,29,21)(16,30,33,42)(17,35,37,44)(18,26,20,24)(22,32,40,34)(23,31,39,36)(28,41,38,43);(1,2)(3,4)(6,10)(11,12)(13,15,19,29)(14,21,25,27)(16,31,33,36)(17,34,37,32)(18,28,20,38)(22,35,40,44)(23,42,39,30)(24,41,26,43);(5,7,9,8)(6,11,10,12)(13,16,19,33)(14,22,25,40)(15,17,29,37)(18,34,20,32)(21,39,27,23)(24,30,26,42)(28,31,38,36)(35,43,44,41);(2,4)(13,17,19,37)(14,23,25,39)(15,16,29,33)(18,36,20,31)(21,40,27,22)(24,35,26,44)(28,32,38,34)(30,43,42,41);(5,8,9,7)(6,11,10,12)(13,18,19,20)(14,24,25,26)(15,28,29,38)(16,32,33,34)(17,36,37,31)(21,41,27,43)(22,42,40,30)(23,35,39,44)

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,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, 7427349);