#482160

order	512
nilpotency class	4
isoclinism	isoclinic class of #481601
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

913 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=i,e^2=h,b^a=be,c^a=cgh,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fgh,f^c=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389356);