#254496

order	512
nilpotency class	3
isoclinism	isoclinic class of #254491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2539	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

863 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=hi,g^2=h,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,d^c=di,e^c=eh,g^c=gh,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161692);