#482773

order	512
nilpotency class	4
isoclinism	isoclinic class of #482734
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	#41985	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#551 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

869 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)(3,7)(4,6)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18)(19,27)(20,24)(21,33,34,25)(22,26)(23,30,35,42)(28,39)(29,40)(31,37)(32,45,44,38)(36,43,46,48)(41,47);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19)(18,23)(20,29)(21,30)(22,28)(24,36)(25,27)(26,35)(31,41)(32,43)(33,39)(34,42)(37,46)(38,40)(44,48)(45,47);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(19,28)(20,31)(23,35)(24,37)(27,39)(30,42)(32,44)(38,45);(3,10)(7,14)(11,16)(13,15)(17,20,22,31)(18,24,26,37)(19,29,28,41)(21,32,34,44)(23,36,35,46)(25,38,33,45)(27,40,39,47)(30,43,42,48)

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fgh,d^c=dhi,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, 389969);