#368874

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1696 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fgi,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,e^d=ei

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, 276070);