#402882

order	512
nilpotency class	3
isoclinism	isoclinic class of #394465
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1157 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)(7,19)(8,23,27,14)(9,15)(10,18)(11,16)(12,17)(13,21)(20,31,32,25)(22,28)(24,30,35,36)(26,33,29,34);(5,7)(6,13)(8,24)(9,16)(10,22)(11,21)(12,15)(14,31)(17,28)(18,19)(20,26)(23,25)(27,35)(29,32)(30,33)(34,36);(5,8)(6,14)(7,20)(9,25)(10,26)(11,29)(12,27)(13,30)(15,32)(16,33)(17,23)(18,34)(19,31)(21,24)(22,35)(28,36);(2,4)(5,9,10,13,12,19,11,28)(6,15,18,21,17,7,16,22)(8,25,26,30,27,31,29,36)(14,32,34,24,23,20,33,35)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250213, #72257150, #72257164
maximal normal subgroup of	#72250213, #72257150, #72257164

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310078);