#7658658

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

3943 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

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, 7565854);