#374287

order	512
nilpotency class	3
isoclinism	isoclinic class of #196110
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3038	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1642 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281483);