#209123

order	512
nilpotency class	3
isoclinism	isoclinic class of #18132
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#120
prime radical	#537	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

779 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=gh,e^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=eh,f^b=fi,d^c=di,e^c=ei

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, 116319);