#405380

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

1300 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=e,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,f^b=fi,g^b=gi,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, 312576);