#362010

order	512
nilpotency class	3
isoclinism	isoclinic class of #24283
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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1382 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 269206);