#347127

order	512
nilpotency class	3
isoclinism	isoclinic class of #347058
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62919	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2221 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,e^b=eh,f^b=fi,e^c=ei,g^d=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254323);