#417078

order	512
nilpotency class	3
isoclinism	isoclinic class of #394801
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

766 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)(9,10,12,13)(11,15,16,14)(18,23)(20,22)(24,27)(25,26)(28,29)(31,32);(1,3,5,4)(2,6,7,8)(10,13)(14,15)(17,18,20,23,21,25,22,26)(19,27,28,24,30,32,29,31);(1,4,5,3)(2,6,7,8)(10,13)(14,15)(17,19,22,28,21,30,20,29)(18,24,26,32,25,31,23,27);(9,11,12,16)(10,14,13,15)(18,25)(23,26)(24,31)(27,32)

Polycyclic presentation:

2^9|a^2=eg,b^2=eh,c^2=ef,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=di,e^c=ei,f^c=fi,g^c=gi

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, 324274);