#414617

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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1502 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=h,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, 321813);