#409974

order	512
nilpotency class	3
isoclinism	isoclinic class of #394026
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,e^c=ei,f^c=fi

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, 317170);