#215923

order	512
nilpotency class	3
isoclinism	isoclinic class of #162948
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

774 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,25,26,23)(20,28)(21,22)(24,29)(27,30,32,31)(33,34)(35,43)(36,42)(37,39)(38,41)(40,45)(44,48)(46,47);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(18,22)(19,26)(20,29)(30,31)(34,39)(35,44)(41,42)(45,46);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31)(33,35,37,44)(34,40,39,47)(36,45,38,46)(41,43,42,48);(2,8)(6,9)(7,16)(14,15)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,36,37,38)(34,41,39,42)(35,45,44,46)(40,43,47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=e,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dhi,f^c=fi,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 123119);