#7642118

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	#36961	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6340 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)(6,8)(7,13)(12,16)(15,19)(25,37)(30,39)(33,47)(36,48)(41,44)(42,49)(46,52)(50,51);(1,2)(3,4)(5,6,10,8)(7,14,13,11)(9,12,17,16)(15,20,19,18)(22,27)(24,35)(28,31)(30,39)(32,45)(36,48)(41,44)(46,52);(5,7,10,13)(6,11,8,14)(9,15,17,19)(12,18,16,20)(21,22,26,27)(23,31,34,28)(24,29,35,38)(25,30,37,39)(32,43,45,40)(33,44,47,41)(36,42,48,49)(46,51,52,50);(5,8,10,6)(7,14,13,11)(9,16,17,12)(15,20,19,18)(21,23,26,34)(22,28,27,31)(24,32,35,45)(25,33,37,47)(29,40,38,43)(30,41,39,44)(36,46,48,52)(42,50,49,51);(5,9,10,17)(6,12,8,16)(7,15,13,19)(11,18,14,20)(21,24)(22,29)(23,32)(25,36)(26,35)(27,38)(28,40)(30,42)(31,43)(33,46)(34,45)(37,48)(39,49)(41,50)(44,51)(47,52);(21,25)(22,30)(23,33)(24,36)(26,37)(27,39)(28,41)(29,42)(31,44)(32,46)(34,47)(35,48)(38,49)(40,50)(43,51)(45,52)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=ehi,d^c=dhi

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, 7549314);