#194959

order	512
nilpotency class	3
isoclinism	isoclinic class of #18446
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 52 points:

(2,4)(5,6,9,13,10,14,8,17)(7,12,15,20,16,19,18,11)(21,22)(23,27)(24,34,36,28)(25,37)(26,30)(29,38)(31,39,44,47)(32,45)(33,41)(35,42,49,48)(40,46)(43,51,52,50);(1,2,3,4)(5,7)(6,11)(8,15)(9,18)(10,16)(12,17)(13,19)(14,20)(21,23,26,33)(22,27,30,41)(24,31,36,44)(25,32,38,46)(28,39,34,47)(29,40,37,45)(35,43,49,52)(42,50,48,51);(5,7)(6,12)(8,18)(9,15)(10,16)(11,17)(13,20)(14,19)(21,24)(22,28)(23,31)(25,35)(26,36)(27,39)(29,42)(30,34)(32,43)(33,44)(37,48)(38,49)(40,50)(41,47)(45,51)(46,52);(5,8,10,9)(6,13,14,17)(7,15,16,18)(11,19,20,12)(21,25,26,38)(22,29,30,37)(23,32,33,46)(24,35,36,49)(27,40,41,45)(28,42,34,48)(31,43,44,52)(39,50,47,51)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102155);