#361676

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	C₂×#3584	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3522 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)(5,6,11,16,12,17,10,15)(7,9,14,27,19,25,20,26)(8,13,24,31,23,30,28,32)(18,33,21,36,34,22,35,29);(2,4)(5,7,10,20,12,19,11,14)(6,9,15,26,17,25,16,27)(8,21,24,34,23,35,28,18)(13,29,31,33,30,36,32,22);(5,8)(6,13)(7,18)(9,22)(10,28)(11,24)(12,23)(14,21)(15,32)(16,31)(17,30)(19,34)(20,35)(25,33)(26,36)(27,29);(2,4)(5,9)(6,14)(7,15)(8,22)(10,26)(11,27)(12,25)(13,21)(16,19)(17,20)(18,32)(23,33)(24,29)(28,36)(30,35)(31,34)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237256, #72239881, #72239887, …
maximal normal subgroup of	#72237256, #72239881, #72239887, …
maximal quotient of	#72237275

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268872);