#502350

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

555 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)(7,11)(8,9)(10,12)(13,14,19,26,17,24,37,43,21,28,18,25,20,27,34,42)(15,22,31,39,16,33,36,41,32,40,30,23,29,38,35,44);(1,2)(3,4)(6,9)(10,11)(13,15,18,30,17,16,34,35,21,32,19,31,20,29,37,36)(14,22,25,23,24,33,42,44,28,40,26,39,27,38,43,41);(2,4)(13,16,21,29)(14,23,28,39)(15,20,32,17)(18,30,19,31)(22,26,40,25)(24,41,27,44)(33,42,38,43)(34,36,37,35);(5,7,8,12)(6,10,9,11)(13,17,21,20)(14,24,28,27)(15,29,32,16)(18,34,19,37)(22,38,40,33)(23,41,39,44)(25,42,26,43)(30,36,31,35)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409546);