#512669

order	512
nilpotency class	5
isoclinism	isoclinic class of #512669
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#127	C₂⁴
prime radical	#60830	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂×#41 ◁ #3498 ◁ G
LECS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,e^2=hi,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=ehi,f^b=fgh,g^b=gi,d^c=di,e^c=ei,f^e=fhi,g^f=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306437, #72318931, #72318941
maximal normal subgroup of	#72306437, #72318931, #72318941

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419865);