#209105

order	512
nilpotency class	3
isoclinism	isoclinic class of #18316
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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

803 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=gh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,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, 116301);