#364186

order	512
nilpotency class	3
isoclinism	isoclinic class of #24278
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1484 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:

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

Polycyclic presentation:

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

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, 271382);