#380561

order	512
nilpotency class	3
isoclinism	isoclinic class of #214688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1467 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,h^b=hi,d^c=de,f^c=fi,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, 287757);