#165016

order	512
nilpotency class	3
isoclinism	isoclinic class of #164187
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2407	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

792 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh,d^c=dghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 72212);