#323839

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

750 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=eghi,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=dh,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, 231035);