#406211

order	512
nilpotency class	3
isoclinism	isoclinic class of #394335
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	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1663 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)(15,25)(16,30)(17,22)(18,23)(19,24)(20,28)(21,33)(26,35)(27,37)(29,36)(31,43)(32,39)(34,38)(40,41)(42,44);(2,4)(5,7,8,11)(6,9,10,12)(13,15)(14,20)(16,31)(17,27)(18,28)(19,29)(21,35)(22,25)(23,36)(24,37)(26,34)(30,40)(32,42)(33,41)(38,44)(39,43);(13,16,18,33,19,34,17,32)(14,21,22,30,24,39,23,38)(15,26,27,41,29,31,28,42)(20,35,36,44,37,43,25,40);(6,10)(9,12)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,32,34,33)(20,25,37,36)(21,30,39,38)(26,41,31,42)(35,40,43,44)

Polycyclic presentation:

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

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, 313407);