#389142

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1168 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=efh,e^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei,h^d=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, 296338);