#359641

order	512
nilpotency class	3
isoclinism	isoclinic class of #188252
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3485	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3908 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256079, #72256257, #72256268
maximal normal subgroup of	#72256079, #72256257, #72256268

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266837);