#303833

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

942 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)(3,7)(4,6)(5,8)(9,10,13,15)(11,12,16,14)(17,18)(19,23)(20,30)(21,24)(22,25)(26,27)(28,32)(29,31);(1,3,5,4)(2,6,8,7)(9,11,13,16)(10,14,15,12)(17,19)(18,23)(20,26)(21,28)(22,29)(24,32)(25,31)(27,30);(1,3,5,4)(2,7,8,6)(9,11,14,10,13,16,12,15)(17,20,30,18,22,24,21,25)(19,26,27,23,29,32,28,31);(1,4,5,3)(2,7,8,6)(9,12,13,14)(10,11,15,16)(17,21,22,30)(18,24,25,20)(19,27,29,28)(23,26,31,32)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=efh,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=dhi

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, 211029);