#205934

order	512
nilpotency class	3
isoclinism	isoclinic class of #205761
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

948 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=f,d^2=i,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=ei,g^b=gi,d^c=dh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113130);