#367105

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1446 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274301);