#411663

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
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	#2408
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1838 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)(7,13)(8,21)(9,17)(10,15)(11,16)(12,19)(14,24)(18,34)(20,30)(22,27)(23,33)(25,32)(26,28)(29,35)(31,36);(5,7)(6,12)(8,18)(9,26)(10,19)(11,20)(13,17)(14,27)(15,30)(16,28)(21,29)(22,25)(23,36)(24,35)(31,32)(33,34);(2,4)(5,7)(6,13)(8,22)(9,26)(10,19)(11,20)(12,15)(14,31)(16,30)(17,28)(18,25)(21,27)(23,35)(24,36)(29,33)(32,34);(5,8,10,24,11,25,9,23)(6,14,17,21,16,33,15,32)(7,18,19,35,20,22,26,36)(12,27,13,29,28,34,30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231521, #72231526, #72231549, …
maximal normal subgroup of	#72231521, #72231526, #72231549, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318859);