#244370

order	512
nilpotency class	3
isoclinism	isoclinic class of #233469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

870 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 870 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=hi,d^2=efhi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=dh,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224779, #72224787
maximal normal subgroup of	#72224779, #72224787

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151566);