#242756

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	C₂²×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

840 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,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, 149952);