#181276

order	512
nilpotency class	3
isoclinism	isoclinic class of #181134
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1240 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,e^2=hi,f^2=h,g^2=hi,b^a=be,e^a=eh,g^a=ghi,f^b=fi,d^c=dhi

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, 88472);