#368653

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	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1080 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:

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

Polycyclic presentation:

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

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, 275849);