#378263

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	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

990 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)(8,10)(9,12)(13,14,20,26)(15,21,29,38)(16,30)(17,36,28,23)(18,24,19,25)(22,34)(27,42)(31,37)(32,44)(33,41)(35,40)(39,43);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,18,28,20,29,19,17)(14,21,24,23,26,38,25,36)(16,31,33,27,35,32,34,43)(22,39,30,37,41,42,40,44);(2,4)(5,7,9,8)(6,11,12,10)(13,16)(14,22)(15,27)(17,32)(18,34)(19,33)(20,35)(21,37)(23,39)(24,40)(25,30)(26,41)(28,31)(29,43)(36,42)(38,44);(5,8,9,7)(6,11,12,10)(13,17,19,29,20,28,18,15)(14,23,25,21,26,36,24,38)(16,32,33,43,35,31,34,27)(22,39,30,37,41,42,40,44)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,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, 285459);