#323334

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

694 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)(17,18,20,25,21,24,28,23)(19,22,26,31,27,30,32,29);(1,3,5,4)(2,6,8,7)(9,10,13,14,12,16,11,15)(17,19,21,27)(18,22,24,30)(20,26,28,32)(23,29,25,31);(10,14)(11,13)(15,16)(18,23)(20,28)(22,29)(24,25)(26,32)(30,31);(1,4,5,3)(2,7,8,6)(18,24)(22,30)(23,25)(29,31)

Polycyclic presentation:

2^9|a^2=ef,b^2=eghi,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230530);