#7522693

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

2137 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)(13,14,20,26)(15,21,29,40)(16,32,28,43)(17,36,27,44)(18,25,19,24)(22,35,39,34)(23,37,38,33)(30,42,31,41);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,21)(16,33)(17,34)(18,31)(19,30)(20,29)(22,36)(23,32)(24,42)(25,41)(26,40)(27,35)(28,37)(38,43)(39,44);(13,16,20,28)(14,22,26,39)(15,27,29,17)(18,35,19,34)(21,38,40,23)(24,43,25,32)(30,33,31,37)(36,42,44,41);(2,4)(5,7,9,8)(6,11,12,10)(13,17,20,27)(14,23,26,38)(15,28,29,16)(18,37,19,33)(21,39,40,22)(24,44,25,36)(30,34,31,35)(32,42,43,41);(5,8,9,7)(6,11,12,10)(15,29)(16,28)(21,40)(22,39)(30,31)(32,43)(34,35)(41,42)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=fi,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, 7429889);