#495224

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

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	C₂²×#508	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1076 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fh,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,d^b=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, 402420);