#497712

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×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

696 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)(7,9)(8,10)(11,12)(14,20)(15,19)(16,24)(17,29)(21,37)(22,39)(23,36)(25,32)(26,41)(27,40)(28,34)(30,33)(35,42)(38,43);(5,7,8,11)(6,9,10,12)(13,14,18,23)(15,21,19,33)(16,26,28,39)(17,30,32,37)(20,25,36,29)(22,40,41,27)(24,43,34,35)(31,38,44,42);(6,10)(9,12)(13,15,18,19)(14,21,23,33)(16,24,28,34)(17,25,32,29)(20,30,36,37)(22,38,41,42)(26,43,39,35)(27,31,40,44);(13,16,18,28)(14,22,23,41)(15,24,19,34)(17,31,32,44)(20,35,36,43)(21,38,33,42)(25,40,29,27)(26,37,39,30)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^2=i,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,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, 404908);