#495228

order	512
nilpotency class	4
isoclinism	isoclinic class of #33618
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#466
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1970
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1073 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fh,e^2=h,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307193, #72307202, #72307277, …
maximal normal subgroup of	#72307193, #72307202, #72307277, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402424);