#209178

order	512
nilpotency class	3
isoclinism	isoclinic class of #18992
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 116374);