#432245

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1389 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 339441);