#121048

order	512
nilpotency class	3
isoclinism	isoclinic class of #121044
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

493 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=hi,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=dh,f^b=fh,e^c=ei,f^c=fi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28244);