#378078

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

915 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=fh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fi,g^c=gi,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, 285274);