#496827

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2184	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

818 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di,f^c=fh,h^c=hi

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, 404023);