#362447

order	512
nilpotency class	3
isoclinism	isoclinic class of #25440
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ 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:

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

Polycyclic presentation:

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

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, 269643);