#8092915

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
prime radical	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4349 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,e^2=gi,f^2=h,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, 8000111);