#368227

order	512
nilpotency class	3
isoclinism	isoclinic class of #23998
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1460 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317818, #72317836, #72317932, …
maximal normal subgroup of	#72317818, #72317836, #72317932, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275423);