#405317

order	512
nilpotency class	3
isoclinism	isoclinic class of #394262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1285 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312513);