#449700

order	512
nilpotency class	3
isoclinism	isoclinic class of #417530
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35150	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1537 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356896);