#495734

order	512
nilpotency class	4
isoclinism	isoclinic class of #33691
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1032 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=gi,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=dh,e^b=ehi,f^b=fi,g^b=gi,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250051, #72250062, #72250096
maximal normal subgroup of	#72250051, #72250062, #72250096
maximal quotient of	#72265509

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402930);