#419784

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1626 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)(7,16,20,17)(8,13)(9,14)(10,26)(11,15)(12,19,28,21)(18,29,33,32)(22,35,34,27)(23,30)(24,36)(25,31);(5,7)(6,12)(8,22)(9,21)(10,19)(11,20)(13,29)(14,17)(15,28)(16,26)(18,23)(24,33)(25,34)(27,30)(31,32)(35,36);(5,6)(7,17)(8,13)(9,14)(10,26)(11,15)(12,19)(16,20)(18,32)(21,28)(22,27)(23,30)(24,36)(25,31)(29,33)(34,35);(2,4)(5,8,9,23,11,25,10,24)(6,13,14,30,15,31,26,36)(7,18,19,22,20,33,21,34)(12,27,16,29,28,35,17,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322924
maximal normal subgroup of	#72322924

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326980);