#165018

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, prime radical	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

671 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=gh,d^2=ghi,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, 72214);