#379509

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	#2487
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

938 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)(7,11)(8,18,20,12)(9,14)(10,13)(15,24,31,35)(16,26)(17,25)(19,27,33,34)(21,30,22,28)(23,29)(32,36);(5,7,9,17,10,16,23,32)(6,11,14,25,13,26,29,36)(8,19,21,31,20,33,22,15)(12,27,28,35,18,34,30,24);(5,8,9,22,10,20,23,21)(6,12,14,30,13,18,29,28)(7,15,17,33,16,31,32,19)(11,24,25,34,26,35,36,27);(1,3,4,2)(5,6)(7,11)(8,12)(9,14)(10,13)(15,24)(16,26)(17,25)(18,20)(19,27)(21,28)(22,30)(23,29)(31,35)(32,36)(33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286705);