#7715673

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#3455	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6358 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622869);