#7520525

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1858 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)(13,14,19,24)(15,20,26,39)(16,28,30,43)(17,32,25,44)(18,33,37,23)(21,35,41,31)(22,36,38,29)(27,42,34,40);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,20)(16,29)(17,31)(18,34)(19,26)(21,32)(22,28)(23,42)(24,39)(25,35)(27,37)(30,36)(33,40)(38,43)(41,44);(5,7,9,8)(6,11,12,10)(13,16)(14,21)(15,25)(17,26)(18,35)(19,30)(20,38)(22,39)(23,43)(24,41)(27,29)(28,33)(31,37)(32,40)(34,36)(42,44);(2,4)(13,17,19,25)(14,22,24,38)(15,16,26,30)(18,36,37,29)(20,21,39,41)(23,44,33,32)(27,35,34,31)(28,40,43,42);(5,8,9,7)(6,11,12,10)(15,26)(16,30)(20,39)(21,41)(27,34)(28,43)(31,35)(40,42)

Polycyclic presentation:

2^9|a^2=i,b^2=f,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, 7427721);