#378151

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

1401 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)(9,10,14,22,16,21,15,20)(11,23,26,37,25,36,24,17)(12,27,29,38,31,18,30,39)(13,32,34,35,33,40,28,19);(1,2,4,3)(5,6)(7,8)(9,11,14,26,16,25,15,24)(10,17,22,23,21,37,20,36)(12,28,29,13,31,34,30,33)(18,32,39,35,27,40,38,19);(6,8)(9,12,16,31)(10,18,21,27)(11,13,25,33)(14,30,15,29)(17,35,37,19)(20,39,22,38)(23,32,36,40)(24,34,26,28);(1,3,4,2)(9,13,16,33)(10,19,21,35)(11,12,25,31)(14,34,15,28)(17,27,37,18)(20,40,22,32)(23,38,36,39)(24,30,26,29)

Polycyclic presentation:

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

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, 285347);