#364194

order	512
nilpotency class	3
isoclinism	isoclinic class of #214672
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1462 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)(7,16,19,11)(8,20,22,13)(9,12)(10,14)(15,24)(17,26)(18,27,23,25)(21,29,34,28)(30,32)(31,35)(33,36);(1,2,4,3)(5,7)(6,11)(8,17)(9,23)(10,19)(12,27)(13,26)(14,16)(15,25)(18,24)(20,30)(21,33)(22,32)(28,36)(29,35)(31,34);(6,12)(9,24)(11,25)(13,28)(14,15)(16,27)(18,23)(20,29)(21,34)(26,35)(30,36)(31,33);(1,3,4,2)(5,8,10,22)(6,13,14,20)(7,17,19,32)(9,21,24,34)(11,26,16,30)(12,28,15,29)(18,31,23,33)(25,35,27,36)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,d^2=egi,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^b=hi,f^c=fi,h^d=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, 271390);