#405801

order	512
nilpotency class	3
isoclinism	isoclinic class of #393900
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	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1890 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)(21,32)(26,42)(27,37)(29,36)(31,41)(33,40)(34,39)(35,44)(38,43);(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)(26,32)(30,42)(33,44)(34,43)(35,39)(40,41);(13,16)(14,21)(15,26)(17,32)(18,33)(19,34)(20,35)(22,39)(23,30)(24,40)(25,41)(27,43)(28,31)(29,44)(36,38)(37,42);(6,10)(9,12)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,32,34,33)(20,25,37,36)(21,39,40,30)(26,43,44,31)(35,41,42,38)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312997);