#242732

order	512
nilpotency class	3
isoclinism	isoclinic class of #242725
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#396
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=h,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,g^a=gh,c^b=cg,g^b=gh,e^c=eh,f^c=fi

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, 149928);