#7562044

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#45245	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2169 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)(13,14,20,26)(15,21,29,22)(16,30,28,38)(17,32,36,41)(18,24,19,25)(23,35,42,34)(27,43,44,40)(31,37,33,39);(1,2)(3,4)(6,10)(11,12)(13,15)(14,21)(16,19)(17,33)(18,28)(20,29)(22,26)(23,40)(24,38)(25,30)(27,35)(31,36)(32,39)(34,44)(37,41)(42,43);(5,7,9,8)(6,11,10,12)(13,16,19,15,20,28,18,29)(14,22,24,38,26,21,25,30)(17,33,35,44,36,31,34,27)(23,37,41,40,42,39,32,43);(2,4)(5,8,9,7)(6,12,10,11)(13,17,18,34,20,36,19,35)(14,23,25,32,26,42,24,41)(15,27,16,31,29,44,28,33)(21,37,38,43,22,39,30,40);(6,10)(11,12)(13,18,20,19)(14,24,26,25)(15,28,29,16)(17,34,36,35)(21,38,22,30)(23,41,42,32)(27,33,44,31)(37,43,39,40)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=fg,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,g^b=gi,h^b=hi,d^c=di,e^c=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, 7469240);