#122178

order	512
nilpotency class	3
isoclinism	isoclinic class of #122176
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1446
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

643 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fh,e^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=gh,d^c=dhi,e^c=eh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29374);