#498685

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

471 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,d^2=ef,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=ci,d^b=de

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405881);