#147098

order	512
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1970
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

536 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=eh,d^2=hi,e^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,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, 54294);