#287725

order	512
nilpotency class	3
isoclinism	isoclinic class of #281915
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	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

652 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:

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

Polycyclic presentation:

2^9|a^2=efhi,b^2=fi,d^2=ei,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^b=gi,d^c=di,f^c=fh,g^c=ghi

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, 194921);