#361618

order	512
nilpotency class	3
isoclinism	isoclinic class of #24278
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×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

3574 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)(9,10,14,20,15,18,13,19)(11,16,23,33,24,31,22,32)(12,25,28,35,29,17,27,36)(21,37,38,34,39,30,26,40);(2,4)(5,6,7,8)(9,11,13,22,15,24,14,23)(10,16,19,32,18,31,20,33)(12,26,27,39,29,38,28,21)(17,34,35,37,25,40,36,30);(6,8)(9,12,15,29)(10,17,18,25)(11,21,24,39)(13,28,14,27)(16,30,31,37)(19,36,20,35)(22,38,23,26)(32,40,33,34);(2,4)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^c=gi,h^c=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, 268814);