#6370370

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2070 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=f,g^2=i,b^a=bf,d^a=df,e^a=ei,c^b=cf,e^b=ei,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277566);