#354484

order	512
nilpotency class	3
isoclinism	isoclinic class of #3019
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1480 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)(7,18,22,31)(8,23)(9,15)(10,16)(11,24)(12,17)(13,21,32,27)(14,25)(19,35,30,29)(20,34,33,36)(26,28);(1,2,3,4)(5,7)(6,13)(8,19)(9,20)(10,27)(11,21)(12,22)(14,29)(15,30)(16,31)(17,32)(18,24)(23,33)(25,34)(26,35)(28,36);(5,8)(6,14)(7,19)(9,16)(10,25)(11,28)(12,15)(13,29)(17,26)(18,33)(20,31)(21,36)(22,30)(23,24)(27,34)(32,35);(5,9,10,26,12,23,11,14)(6,15,16,28,17,8,24,25)(7,20,27,35,22,33,21,29)(13,30,31,36,32,19,18,34)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250386, #72252972, #72256886
maximal normal subgroup of	#72250386, #72252972, #72256886

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261680);