#394775

order	512
nilpotency class	3
isoclinism	isoclinic class of #394020
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1630 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)(7,14)(8,20)(9,15)(10,25)(11,16)(12,18)(13,23)(17,30)(19,28)(21,31)(22,33)(24,32)(26,29)(27,35)(34,36);(1,2,4,3)(5,7)(6,12)(8,21)(9,18)(10,19)(11,26)(13,30)(14,15)(16,28)(17,24)(20,27)(22,35)(23,34)(25,29)(31,33)(32,36);(1,3,4,2)(5,8,9,22)(6,13,15,32)(7,17,18,34)(10,24,11,23)(12,27,14,31)(16,33,25,20)(19,35,26,21)(28,30,29,36);(5,7)(6,14)(8,17)(9,18)(10,26)(11,19)(12,15)(13,31)(16,28)(20,36)(21,24)(22,34)(23,35)(25,29)(27,32)(30,33)

Polycyclic presentation:

2^9|b^2=ei,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gi,f^c=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, 301971);