#362963

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1054 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)(13,14,19,24,20,26,18,25)(15,27,29,37,30,23,17,21)(16,31,33,22,35,39,34,40)(28,42,32,41,44,36,43,38);(5,6,8,10)(7,11,12,9)(13,15,19,29,20,30,18,17)(14,21,25,23,26,37,24,27)(16,28,34,43,35,44,33,32)(22,36,39,38,40,42,31,41);(2,4)(5,7,8,12)(6,9,10,11)(13,16)(14,22)(15,28)(17,32)(18,33)(19,34)(20,35)(21,36)(23,38)(24,31)(25,39)(26,40)(27,41)(29,43)(30,44)(37,42);(13,17,18,30,20,29,19,15)(14,23,24,21,26,27,25,37)(16,32,33,44,35,43,34,28)(22,38,31,36,40,41,39,42)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270159);