#371861

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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

1074 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 279057);