#7857901

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	#39865	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4544 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7765097);