#124166

order	512
nilpotency class	4
isoclinism	isoclinic class of #1339
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#327
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×#97	C₂³
socle	C₂	#7565
prime radical	#3054	C₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

371 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=h,d^2=gi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,d^a=df,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31362);