#378212

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

836 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)(7,11)(8,12)(9,10)(13,14)(15,27)(16,31,34,41)(17,23)(18,24)(19,25)(20,26)(21,29)(22,35,40,36)(28,39,43,44)(30,38)(32,42,33,37);(1,2)(3,4)(6,10)(11,12)(13,15,19,30,18,29,20,17)(14,21,25,23,24,27,26,38)(16,32,35,28,34,33,36,43)(22,39,31,37,40,44,41,42);(2,4)(5,7,9,8)(6,11,10,12)(13,16,18,34)(14,22,24,40)(15,28,29,43)(17,33,30,32)(19,36,20,35)(21,37,27,42)(23,39,38,44)(25,41,26,31);(5,8,9,7)(6,11,10,12)(13,17,20,29,18,30,19,15)(14,23,26,21,24,38,25,27)(16,33,35,43,34,32,36,28)(22,39,31,37,40,44,41,42)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,d^2=efh,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, 285408);