#244118

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

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 706 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:

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

Polycyclic presentation:

2^9|a^2=eh,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	#72283798, #72283823, #72283833, …
maximal normal subgroup of	#72283798, #72283823, #72283833, …
maximal quotient of	C₂×#244118

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151314);