#391846

order	512
nilpotency class	3
isoclinism	isoclinic class of #213798
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1034 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:

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

Polycyclic presentation:

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

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, 299042);