#405861

order	512
nilpotency class	3
isoclinism	isoclinic class of #395781
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1000 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313057);