#283827

order	512
nilpotency class	3
isoclinism	isoclinic class of #283771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

772 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 191023);