#449751

order	512
nilpotency class	3
isoclinism	isoclinic class of #417700
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	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1425 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 356947);