#497021

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

594 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fhi,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,d^c=di

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, 404217);