#419062

order	512
nilpotency class	3
isoclinism	isoclinic class of #417508
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#35347	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2749 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2,3,4)(5,6)(7,8)(9,10,14,13)(11,15,19,18)(12,20,22,16)(17,24,21,23);(9,11)(10,15)(12,21,22,17)(13,18)(14,19)(16,23,20,24);(2,4)(9,12)(10,16)(11,17)(13,20)(14,22)(15,23)(18,24)(19,21);(6,8)(9,13,14,10)(11,18,19,15)(12,20,22,16)(17,24,21,23)

Polycyclic presentation:

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

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, 326258);