#346784

order	512
nilpotency class	3
isoclinism	isoclinic class of #346740
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3554	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2215 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(6,11)(8,9)(12,18)(13,17)(14,16)(19,20)(21,22,26,30)(23,31,32,35)(24,27,25,29)(28,33,36,34);(5,6)(7,12)(8,11)(9,17)(10,13)(14,19)(15,20)(16,18)(22,27)(24,25)(28,35)(29,30)(31,36)(33,34);(5,7,10,15)(6,12,13,20)(8,16,9,14)(11,18,17,19)(21,23,26,32)(22,28,30,36)(24,34,25,33)(27,35,29,31);(2,4)(5,8,10,9)(6,11,13,17)(7,14,15,16)(12,19,20,18)(23,32)(28,36)(31,35)(33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253980);