#276122

order	512
nilpotency class	3
isoclinism	isoclinic class of #267291
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2867	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

706 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)(4,5)(6,8)(9,10,14,16,12,11,13,15)(17,18,20,24,21,25,30,23)(19,26,27,32,28,22,29,31);(2,5)(6,7)(10,11)(15,16)(17,19,21,28)(18,22,25,26)(20,27,30,29)(23,32,24,31);(9,11,12,10)(13,15,14,16)(18,23)(19,27)(20,30)(24,25)(28,29)(31,32);(1,3,4,8)(2,6,5,7)(17,20,21,30)(18,24,25,23)(19,27,28,29)(22,31,26,32)

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=hi,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 183318);