#504357

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

574 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=fi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=eh,h^b=hi,e^d=eh,h^d=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, 411553);