#347316

order	512
nilpotency class	3
isoclinism	isoclinic class of #347226
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	C₂×C₄²	C₂⁴
prime radical	#63000	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2175 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268185
maximal normal subgroup of	#72268185

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254512);