#6400648

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1649 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=fi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6307844);