#7734375

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	#47849	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4487 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(2,6)(7,8)(9,10)(11,18)(12,13)(14,16)(15,17)(19,24)(20,21)(22,23)(25,26,30,29)(27,32,35,31)(28,33,38,37)(34,40,36,39);(1,2)(3,7)(4,8)(5,6)(9,10,15,17)(11,18,12,13)(14,22,23,16)(19,21,20,24)(25,27)(26,31)(28,34)(29,32)(30,35)(33,39)(36,38)(37,40);(1,3,5,4)(2,7,6,8)(9,11,15,12)(10,13,17,18)(14,19,23,20)(16,21,22,24)(25,27,30,35)(26,32,29,31)(28,36,38,34)(33,39,37,40);(1,3,5,4)(2,7,6,8)(9,12,15,11)(10,13,17,18)(14,20,23,19)(16,21,22,24)(25,28,30,38)(26,33,29,37)(27,34,35,36)(31,39,32,40);(9,13)(10,11)(12,17)(14,21)(15,18)(16,19)(20,22)(23,24)(25,27)(26,32)(28,34)(29,31)(30,35)(33,40)(36,38)(37,39);(1,4,5,3)(2,8,6,7)(9,14,15,23)(10,16,17,22)(11,19,12,20)(13,21,18,24)(25,29,30,26)(27,31,35,32)(28,37,38,33)(34,39,36,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7641571);