#364428

order	512
nilpotency class	3
isoclinism	isoclinic class of #196161
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2470 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271624);