#391606

order	512
nilpotency class	3
isoclinism	isoclinic class of #232181
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

748 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298802);