#7714924

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

3995 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:

(2,6)(7,8)(10,12)(15,17)(16,20)(22,23)(29,41)(34,43)(37,48)(39,52)(45,53)(46,51)(50,55)(54,56);(1,2)(3,7)(4,8)(5,6)(9,10,14,12)(11,15,19,17)(13,16,21,20)(18,22,24,23)(26,31)(27,35)(33,38)(34,43)(36,47)(37,48)(46,51)(50,55);(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,26)(27,32)(28,38)(29,34)(30,31)(33,40)(35,42)(36,49)(37,45)(39,51)(41,43)(44,47)(46,52)(48,53)(50,56)(54,55);(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27,30,35)(26,32,31,42)(28,36,40,47)(29,37,41,48)(33,44,38,49)(34,45,43,53)(39,50,52,55)(46,54,51,56);(1,3,5,4)(2,7,6,8)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(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);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(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|b^2=h,c^2=h,d^2=hi,e^2=gi,f^2=g,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, 7622120);