#244117

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2397
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

792 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,3,6,4,7,8,5)(9,10,13,17,14,18,23,29)(11,19,21,27,22,32,26,15)(12,16,24,30,25,31,36,39)(20,33,34,38,35,40,37,28);(2,5)(3,8)(6,7)(9,11)(10,15)(12,20)(13,26)(14,22)(16,28)(17,32)(18,27)(19,29)(21,23)(24,37)(25,35)(30,40)(31,38)(33,39)(34,36);(2,5)(3,8)(6,7)(15,27)(19,32)(28,38)(33,40);(9,12,23,36,14,25,13,24)(10,16,29,39,18,31,17,30)(11,20,21,34,22,35,26,37)(15,28,19,33,27,38,32,40)

Polycyclic presentation:

2^9|a^2=e,d^2=efh,e^2=h,f^2=i,b^a=be,c^a=cf,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,e^c=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283823, #72283827, #72283833, …
maximal normal subgroup of	#72283823, #72283827, #72283833, …
maximal quotient of	C₂×#244117

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151313);