#7614448

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	#45269	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1844 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)(13,14,18,22)(15,23,28,34)(16,29,32,44)(17,33,25,21)(19,24,36,27)(20,37,39,30)(26,38,31,35)(40,41,43,42);(2,4)(6,10)(11,12)(15,24)(16,30)(17,25)(19,34)(20,29)(21,33)(23,36)(27,28)(32,37)(39,44)(40,43)(41,42);(5,7,9,8)(6,11,10,12)(13,15,25,24,18,28,17,27)(14,19,33,34,22,36,21,23)(16,31,37,42,32,26,30,41)(20,38,44,40,39,35,29,43);(1,2,3,4)(13,16,17,30,18,32,25,37)(14,20,21,29,22,39,33,44)(15,26,27,42,28,31,24,41)(19,35,23,40,36,38,34,43);(5,8,9,7)(6,11,10,12)(16,32)(20,39)(26,31)(29,44)(30,37)(35,38)(40,43)(41,42)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=h,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,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7521644);