#8088330

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	#44231	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4502 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=ghi,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7995526);