#365065

order	512
nilpotency class	3
isoclinism	isoclinic class of #24295
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3058	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1732 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)(7,10)(8,17)(9,24)(11,15)(12,16)(13,22)(14,25)(18,27)(19,21)(20,28)(23,29)(26,30)(31,33)(32,35)(34,36);(1,3,4,2)(5,7,11,16,12,15,10,6)(8,19,22,29,23,13,21,17)(9,18,26,30,27,24,25,14)(20,32,34,36,35,28,33,31);(5,8)(6,13)(7,17)(9,20)(10,21)(11,22)(12,23)(14,28)(15,29)(16,19)(18,31)(24,36)(25,33)(26,34)(27,35)(30,32);(5,9,11,26,12,27,10,25)(6,14,7,18,16,30,15,24)(8,20,22,34,23,35,21,33)(13,28,17,31,19,32,29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312212, #72312257, #72312295
maximal normal subgroup of	#72312212, #72312257, #72312295

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272261);