#498007

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	C₂²×D₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

598 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,d^c=di,e^d=eh,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405203);