#368460

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1414 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)(9,10,14,19,15,20,13,18)(11,16,23,32,24,33,22,31)(12,25,27,17,29,35,28,36)(21,34,37,30,38,40,26,39);(2,4)(5,7,8,6)(9,11)(10,16)(12,26)(13,22)(14,23)(15,24)(17,34)(18,31)(19,32)(20,33)(21,27)(25,39)(28,38)(29,37)(30,35)(36,40);(9,12,13,27,15,29,14,28)(10,17,18,35,20,36,19,25)(11,21,23,26,24,38,22,37)(16,30,32,34,33,39,31,40);(2,4)(9,13,15,14)(10,18,20,19)(11,22,24,23)(12,27,29,28)(16,31,33,32)(17,35,36,25)(21,37,38,26)(30,40,39,34)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275656);