#8093076

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4481 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)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,13,15)(12,16,21,24)(19,23,20,22)(25,26,30,34)(27,35,37,31)(28,33,29,32)(36,39,38,40);(1,3)(2,5)(4,7)(6,8)(9,11,14,13)(10,15,17,18)(12,19,21,20)(16,22,24,23)(25,27,30,37)(26,31,34,35)(28,36,29,38)(32,39,33,40);(2,6)(3,7)(9,12,14,21)(10,16,17,24)(11,19,13,20)(15,22,18,23)(25,28,30,29)(26,32,34,33)(27,36,37,38)(31,39,35,40);(3,7)(5,8)(9,11,14,13)(10,15,17,18)(12,20,21,19)(16,23,24,22)(25,29,30,28)(26,33,34,32)(27,36,37,38)(31,39,35,40);(10,17)(15,18)(16,24)(22,23);(9,13,14,11)(10,15,17,18)(12,20,21,19)(16,22,24,23)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^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, 8000272);