#7627627

order	512
nilpotency class	2
isoclinism	isoclinic class of #62972
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6314 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fg,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=eh

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534823);