#449593

order	512
nilpotency class	3
isoclinism	isoclinic class of #417566
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3020	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1997 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)(7,15,18,11)(8,12)(9,14)(10,13)(16,19,29,23)(17,31,34,26)(20,28)(21,27)(22,25)(24,30)(32,36,33,35);(5,6)(7,16)(8,12)(9,22)(10,13)(11,19)(14,25)(15,23)(17,33)(18,29)(20,30)(21,27)(24,28)(26,35)(31,36)(32,34);(5,7)(6,11)(8,17)(9,23)(10,18)(12,26)(13,15)(14,29)(16,22)(19,25)(20,36)(21,34)(24,35)(27,31)(28,32)(30,33);(2,4)(5,8,10,21)(6,12,13,27)(7,17,18,34)(9,24,25,20)(11,26,15,31)(14,30,22,28)(16,32,29,33)(19,36,23,35)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356789);