#364511

order	512
nilpotency class	3
isoclinism	isoclinic class of #23998
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂²×#454	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2416 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)(3,4)(5,6)(7,10)(8,18,21,29)(9,16)(11,14)(12,20,15,25)(13,22)(17,24)(19,32,34,36)(23,30)(26,28)(27,35,31,33);(2,4)(5,7,10,6)(8,12)(9,16,22,13)(11,17,24,14)(15,21)(18,20)(19,27)(23,30,26,28)(25,29)(31,34)(32,35)(33,36);(5,8)(6,12)(7,15)(9,19)(10,21)(11,25)(13,27)(14,29)(16,31)(17,18)(20,24)(22,34)(23,35)(26,33)(28,32)(30,36);(2,4)(5,9,10,22)(6,13,7,16)(8,19,21,34)(11,26,24,23)(12,27,15,31)(14,30,17,28)(18,32,29,36)(20,35,25,33)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,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,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317821, #72317836, #72317937, …
maximal normal subgroup of	#72317821, #72317836, #72317937, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271707);