#242760

order	512
nilpotency class	3
isoclinism	isoclinic class of #242755
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

652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,11)(4,9)(5,8)(6,10)(7,14)(12,15)(13,16)(17,18,21,26,22,25,31,32)(19,23,27,30,24,20,28,29);(1,3,4,14,6,12,5,13)(2,7,8,11,10,16,9,15)(17,19)(18,23)(20,25)(21,27)(22,24)(26,30)(28,31)(29,32);(2,8)(3,12)(4,5)(7,11)(9,10)(15,16)(17,20,22,23)(18,24,25,19)(21,30,31,29)(26,27,32,28);(1,4,6,5)(2,8,10,9)(3,13,12,14)(7,15,16,11)(18,25)(19,24)(26,32)(27,28)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=hi,d^2=h,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,d^b=di,g^b=gh,d^c=dh,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, 149956);