#367902

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1187 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275098);