#433903

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:

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

Polycyclic presentation:

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

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, 341099);