#481959

order	512
nilpotency class	4
isoclinism	isoclinic class of #11799
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
socle	C₂²	C₂²×#99
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

863 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)(3,11,13,16)(4,9)(5,10)(6,8)(7,12,15,14)(17,18)(19,27)(20,24)(21,34,35,25)(22,26)(23,29,37,44)(28,41)(30,42)(31,45)(32,47,48,39)(33,40)(36,43,38,46);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,22,30)(18,23,26,37)(20,31,33,28)(21,29,35,44)(24,38,40,36)(25,27,34,42)(32,46,48,43)(39,45,47,41);(2,8)(7,15)(9,10)(11,16)(17,20,22,33)(18,24,26,40)(19,28,30,31)(21,32,35,48)(23,36,37,38)(25,39,34,47)(27,41,42,45)(29,43,44,46);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389155);