#195304

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1114 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)(5,6,10,14,11,7,9,13)(8,16,19,12,17,20,18,15)(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)(5,7,11,6)(8,15,17,12)(9,13,10,14)(16,19,20,18)(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);(5,8,11,17)(6,12,7,15)(9,18,10,19)(13,16,14,20)(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);(8,17)(12,15)(16,20)(18,19)(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,c^2=h,d^2=i,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=di,f^a=fhi,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fh,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, 102500);