#165128

order	512
nilpotency class	3
isoclinism	isoclinic class of #164512
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

659 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)(3,11)(4,9)(5,8)(6,10)(7,12)(13,15)(14,16)(18,22,24,27)(19,25)(23,30,31,29)(26,32)(33,34,37,42,38,40,36,41)(35,43,45,39,46,48,44,47);(1,3,6,13)(2,7,10,16)(4,12,5,14)(8,11,9,15)(17,18,21,24)(19,23,25,31)(20,27,28,22)(26,30,32,29)(33,35,38,46)(34,39,40,47)(36,45,37,44)(41,48,42,43);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,11,16,15)(17,19)(18,23)(20,26)(21,25)(22,29)(24,31)(27,30)(28,32)(34,40)(39,47)(41,42)(43,48);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,15,16,11)(19,25)(23,31)(26,32)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 72324);