#397803

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

1054 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=e,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^c=fi,g^c=gi,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, 304999);