#366735

order	512
nilpotency class	3
isoclinism	isoclinic class of #25196
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#454	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2213 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)(8,19)(9,24)(10,18)(11,14)(12,22)(13,25)(16,23)(17,27)(20,33)(21,29)(26,30)(28,35)(31,36)(32,34);(5,7,10,6,11,18,15,14)(8,19,22,12,23,16,21,29)(9,17,26,13,27,24,25,30)(20,32,34,28,35,31,36,33);(2,4)(5,8)(6,12)(7,16)(9,20)(10,21)(11,23)(13,28)(14,29)(15,22)(17,31)(18,19)(24,32)(25,34)(26,36)(27,35)(30,33);(5,9,11,27)(6,13,14,30)(7,17,18,24)(8,20,23,35)(10,26,15,25)(12,28,29,33)(16,31,19,32)(21,36,22,34)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273931);