#495341

order	512
nilpotency class	3
isoclinism	isoclinic class of #11234
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#391
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1985
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

761 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,c^b=ch,f^b=fi,g^b=gi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239242, #72249769, #72249773, …
maximal normal subgroup of	#72239242, #72249769, #72249773, …
maximal quotient of	#72239286, #72323001

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402537);