#7644282

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4047 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 56 points:

(2,5)(6,8)(9,10,14,13)(11,17,18,15)(12,20,21,16)(19,22,24,23)(29,41)(34,43)(37,48)(39,52)(45,53)(46,51)(50,55)(54,56);(1,2,4,5)(3,6,7,8)(11,18)(15,17)(19,24)(22,23)(26,31)(27,35)(33,38)(34,43)(36,47)(37,48)(46,51)(50,55);(9,11,14,18)(10,15,13,17)(12,19,21,24)(16,22,20,23)(25,26,30,31)(27,32,35,42)(28,38,40,33)(29,34,41,43)(36,49,47,44)(37,45,48,53)(39,51,52,46)(50,56,55,54);(1,3,4,7)(2,6,5,8)(9,12,14,21)(10,16,13,20)(11,19,18,24)(15,22,17,23)(25,27)(26,32)(28,36)(29,37)(30,35)(31,42)(33,44)(34,45)(38,49)(39,50)(40,47)(41,48)(43,53)(46,54)(51,56)(52,55);(9,10,14,13)(11,15,18,17)(12,16,21,20)(19,22,24,23)(25,28,30,40)(26,33,31,38)(27,36,35,47)(29,39,41,52)(32,44,42,49)(34,46,43,51)(37,50,48,55)(45,54,53,56);(9,13,14,10)(11,17,18,15)(12,20,21,16)(19,23,24,22)(25,29)(26,34)(27,37)(28,39)(30,41)(31,43)(32,45)(33,46)(35,48)(36,50)(38,51)(40,52)(42,53)(44,54)(47,55)(49,56)

Polycyclic presentation:

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

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, 7551478);