#248734

order	512
nilpotency class	3
isoclinism	isoclinic class of #248647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1324 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155930);