#241400

order	512
nilpotency class	3
isoclinism	isoclinic class of #234079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 148596);