#495035

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2)(3,4)(5,6,10,16,12,17,11,32)(7,13,21,36,19,34,22,50)(8,23,27,53,25,51,28,56)(9,29,30,40,31,57,24,15)(14,37,39,55,38,54,52,26)(18,41,44,63,46,64,45,67)(20,48,49,61,47,68,42,35)(33,58,59,66,60,65,62,43);(2,4)(5,7,10,21,12,19,11,22)(6,13,16,36,17,34,32,50)(8,18,27,44,25,46,28,45)(9,20,30,49,31,47,24,42)(14,33,39,59,38,60,52,62)(15,35,29,48,40,61,57,68)(23,41,53,63,51,64,56,67)(26,43,37,58,55,66,54,65);(5,8,24,54,11,28,31,55,12,25,30,37,10,27,9,26)(6,14,29,53,16,39,40,51,17,38,57,56,32,52,15,23)(7,18,42,65,22,45,47,66,19,46,49,58,21,44,20,43)(13,33,48,63,36,59,61,64,34,60,68,67,50,62,35,41);(7,19)(8,25)(13,34)(14,38)(20,47)(21,22)(23,51)(26,55)(27,28)(35,61)(36,50)(37,54)(39,52)(42,49)(48,68)(53,56)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402231);