#368663

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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

1074 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efhi,c^2=g,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^c=fi,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, 275859);