#384517

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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

934 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)(7,10)(8,12)(9,11)(13,14,20,26,19,25,18,24)(15,27,31,21,30,40,29,39)(16,32,35,23,17,22,34,42)(28,41,33,43,44,38,36,37);(5,7,9,8)(6,10,11,12)(13,15,19,30)(14,21,25,39)(16,33,17,36)(18,31,20,29)(22,41,32,38)(23,43,42,37)(24,40,26,27)(28,34,44,35);(5,8,9,7)(6,10,11,12)(13,16,19,17)(14,22,25,32)(15,28,30,44)(18,34,20,35)(21,37,39,43)(23,26,42,24)(27,38,40,41)(29,36,31,33);(6,11)(10,12)(13,17,19,16)(14,23,25,42)(15,28,30,44)(18,34,20,35)(21,38,39,41)(22,26,32,24)(27,37,40,43)(29,33,31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 291713);