#480407

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 ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ 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)(11,19,20,34)(12,16)(13,26,28,17)(14,18)(15,22)(21,35,23,37)(24,39,40,32)(25,33)(27,29)(30,36)(31,38);(9,11)(10,15)(12,23)(13,27)(14,20)(16,31)(17,34)(18,29)(19,26)(21,25)(22,28)(24,36)(30,33)(32,35)(37,39)(38,40);(2,6)(7,8)(11,20)(15,29)(19,34)(21,23)(22,27)(30,31)(35,37)(36,38);(1,3,5,4)(2,7,6,8)(9,12,14,25)(10,16,18,33)(11,21,20,23)(13,24,28,40)(15,30,29,31)(17,32,26,39)(19,35,34,37)(22,36,27,38)

Polycyclic presentation:

2^9|a^2=f,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, 387603);