#397015

order	512
nilpotency class	3
isoclinism	isoclinic class of #395148
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1037 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 304211);