#371121

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	#2580
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

908 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)(7,15,18,11)(8,12)(9,14)(10,13)(16,19,28,22)(17,24,31,29)(20,26)(21,25)(23,27)(30,34,35,32)(33,36);(1,3,4,2)(5,6)(7,16,18,28)(8,12)(9,14)(10,13)(11,19,15,22)(17,30,31,35)(20,26)(21,25)(23,27)(24,32,29,34)(33,36);(5,7)(6,11)(8,17)(9,22)(10,18)(12,24)(13,15)(14,28)(16,27)(19,23)(20,31)(21,34)(25,35)(26,29)(30,36)(32,33);(5,8,9,21,10,20,23,33)(6,12,14,25,13,26,27,36)(7,17,22,34,18,31,19,32)(11,24,28,35,15,29,16,30)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=eg,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, 278317);