#7715494

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6120 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)(6,10)(8,16)(11,13)(14,20)(21,22,27,24)(23,31,32,28)(25,35,39,29)(26,36,40,30)(33,41,37,43)(34,42,46,45)(38,44,50,48)(47,52,49,51);(1,2)(3,4)(22,24)(23,32)(29,35)(30,36)(33,37)(34,46)(44,48)(47,49);(5,6,9,10)(7,13,15,11)(8,12,16,17)(14,19,20,18)(21,23,27,32)(22,28,24,31)(25,37,39,33)(26,34,40,46)(29,43,35,41)(30,42,36,45)(38,49,50,47)(44,52,48,51);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19)(21,24,27,22)(23,28,32,31)(25,35,39,29)(26,36,40,30)(33,41,37,43)(34,42,46,45)(38,48,50,44)(47,51,49,52);(5,8)(6,12)(7,14)(9,16)(10,17)(11,18)(13,19)(15,20)(21,25)(22,29)(23,33)(24,35)(26,38)(27,39)(28,41)(30,44)(31,43)(32,37)(34,47)(36,48)(40,50)(42,51)(45,52)(46,49);(21,26,27,40)(22,30,24,36)(23,34,32,46)(25,38,39,50)(28,42,31,45)(29,44,35,48)(33,47,37,49)(41,51,43,52)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=h,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, 7622690);