#480411

order	512
nilpotency class	4
isoclinism	isoclinic class of #480263
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1005 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=hi,e^2=i,f^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,c^b=ci,d^b=di,e^b=ei,f^b=fgi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387607);