#6370272

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2040 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=f,g^2=i,b^a=bf,d^a=df,c^b=cfi,e^b=ei,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277468);