#241236

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#120
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

772 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148432);