#433910

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1235 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)(5,6,8,10)(7,9,11,12)(13,14)(15,22)(16,30)(17,32)(18,25)(19,24)(20,26)(21,28)(23,34)(27,42)(29,37)(31,39)(33,41)(35,40)(36,44)(38,43);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,28,20,29,19,16)(14,21,24,22,26,30,25,37)(17,31,33,44,35,43,34,27)(23,39,40,42,41,38,32,36);(7,11)(9,12)(13,16,19,29,20,28,18,15)(14,22,25,21,26,37,24,30)(17,31,33,44,35,43,34,27)(23,38,40,36,41,39,32,42);(2,4)(13,17)(14,23)(15,27)(16,31)(18,34)(19,33)(20,35)(21,36)(22,38)(24,32)(25,40)(26,41)(28,43)(29,44)(30,42)(37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=fi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,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, 341106);