#181060

order	512
nilpotency class	3
isoclinism	isoclinic class of #17386
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	C₂³	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1310 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=i,g^2=h,b^a=be,e^a=eh,c^b=ci,f^b=fh,d^c=dh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88256);