#323843

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

626 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=efh,b^2=eghi,c^2=h,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, 231039);