#7715504

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

6262 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)(6,7)(9,17)(11,13)(15,20)(21,22,27,28)(23,25,30,38)(24,29,34,41)(26,39,40,31)(32,35,42,47)(33,46,44,37)(36,50,48,43)(45,52,51,49);(1,2)(3,4)(5,6,10,7)(8,11,16,13)(9,12,17,14)(15,18,20,19)(22,28)(23,30)(24,34)(31,39)(33,44)(35,47)(36,48)(49,52);(5,7,10,6)(8,11,16,13)(9,14,17,12)(15,18,20,19)(21,23)(22,25)(24,32)(26,33)(27,30)(28,38)(29,35)(31,37)(34,42)(36,45)(39,46)(40,44)(41,47)(43,49)(48,51)(50,52);(5,8,10,16)(6,11,7,13)(9,15,17,20)(12,18,14,19)(21,24)(22,29)(23,32)(25,35)(26,36)(27,34)(28,41)(30,42)(31,43)(33,45)(37,49)(38,47)(39,50)(40,48)(44,51)(46,52);(5,9)(6,12)(7,14)(8,15)(10,17)(11,18)(13,19)(16,20)(21,25)(22,30)(23,28)(24,35)(26,37)(27,38)(29,42)(31,44)(32,41)(33,39)(34,47)(36,49)(40,46)(43,51)(45,50)(48,52);(21,26,27,40)(22,31,28,39)(23,33,30,44)(24,36,34,48)(25,37,38,46)(29,43,41,50)(32,45,42,51)(35,49,47,52)

Polycyclic presentation:

2^9|a^2=i,b^2=h,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, 7622700);