#301953

order	512
nilpotency class	3
isoclinism	isoclinic class of #301949
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62272	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1148 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=eh,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 209149);