#181714

order	512
nilpotency class	3
isoclinism	isoclinic class of #17597
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,9)(4,12)(5,8)(6,11)(7,13)(10,15)(14,16)(17,18,21,26,23,27,22,36)(19,28,31,42,30,24,32,38)(20,25,34,40,33,39,35,46)(29,41,44,48,43,37,45,47);(1,3)(2,6,8,14)(4,11)(5,10)(7,9,12,15)(13,16)(17,19,22,32,23,30,21,31)(18,24,26,38,27,28,36,42)(20,29,35,45,33,43,34,44)(25,37,40,47,39,41,46,48);(3,10)(6,14)(9,15)(11,16)(17,20)(18,25)(19,29)(21,34)(22,35)(23,33)(24,37)(26,40)(27,39)(28,41)(30,43)(31,44)(32,45)(36,46)(38,47)(42,48);(20,33)(25,39)(29,43)(34,35)(37,41)(40,46)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=h,f^2=i,g^2=i,b^a=be,e^a=eh,g^a=ghi,c^b=ch,e^b=ei,d^c=di

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88910);