#7515115

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1934 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)(13,14,19,24)(15,20,23,18)(16,25,30,36)(17,26,33,38)(21,29,27,31)(22,34,37,32)(28,35,43,40)(39,42,41,44);(1,2)(3,4)(6,9)(10,12)(13,15,19,23)(14,18,24,20)(16,27,30,21)(17,22,33,37)(25,29,36,31)(26,32,38,34)(28,41,43,39)(35,42,40,44);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,25)(17,28)(18,29)(19,30)(20,31)(22,35)(23,36)(24,27)(26,39)(32,42)(33,43)(34,44)(37,40)(38,41);(2,4)(13,17,19,33)(14,22,24,37)(15,26,23,38)(16,28,30,43)(18,34,20,32)(21,35,27,40)(25,39,36,41)(29,44,31,42);(6,9)(10,12)(13,18,19,20)(14,23,24,15)(16,29,30,31)(17,32,33,34)(21,36,27,25)(22,26,37,38)(28,42,43,44)(35,39,40,41)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,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,g^c=gi,h^c=hi,e^d=ei,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, 7422311);