#368895

order	512
nilpotency class	3
isoclinism	isoclinic class of #222240
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1730 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314443, #72314460, #72314471
maximal normal subgroup of	#72314443, #72314460, #72314471
maximal quotient of	#72314517

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276091);