#8093101

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4666 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,21)(13,17)(14,15)(18,22)(19,24)(20,23)(25,26)(27,34)(28,38)(29,32)(30,33)(31,37)(35,39)(36,40);(3,4)(6,8)(10,15)(16,22)(17,21)(23,24)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,19)(13,20)(14,18)(15,22)(17,23)(21,24)(25,28,30,29)(26,32,33,38)(27,35,37,36)(31,39,34,40);(1,4,5,3)(2,6,7,8)(10,15)(11,18)(17,21)(19,20)(25,28,30,29)(26,32,33,38)(27,36,37,35)(31,40,34,39);(9,12,14,13)(10,17,15,21)(11,19,18,20)(16,23,22,24)(25,29,30,28)(26,32,33,38)(27,36,37,35)(31,39,34,40);(9,13,14,12)(10,17,15,21)(11,20,18,19)(16,23,22,24)

Polycyclic presentation:

2^9|c^2=h,d^2=h,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,d^c=dghi

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, 8000297);