#358789

order	512
nilpotency class	3
isoclinism	isoclinic class of #2032
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator	C₂⁴	C₂³×C₄
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
socle	C₂²	C₂×#393
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2710 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)(7,11,17,24)(8,18)(9,13)(10,14)(12,20)(15,25,29,22)(16,23,30,34)(19,31)(21,27)(26,32)(28,35,36,33);(2,4)(5,7)(6,11,14,24)(8,15)(9,16)(10,17)(12,25,18,22)(13,23,27,34)(19,28)(20,29)(21,30)(26,35,31,33)(32,36);(5,8)(6,12,14,18)(7,15)(9,19)(10,20)(11,22,24,25)(13,26,27,31)(16,28)(17,29)(21,32)(23,33,34,35)(30,36);(5,9,10,21)(6,13,14,27)(7,16,17,30)(8,19,20,32)(11,23,24,34)(12,26,18,31)(15,28,29,36)(22,33,25,35)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237658, #72237660, #72237666, …
maximal normal subgroup of	#72237658, #72237660, #72237666, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265985);