#7521996

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

2095 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=f,e^2=fi,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, 7429192);