#205998

order	512
nilpotency class	3
isoclinism	isoclinic class of #205795
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1137 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=fh,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=ei,g^b=gi,d^c=dhi

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, 113194);