#120307

order	512
nilpotency class	3
isoclinism	isoclinic class of #10518
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1441
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

538 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=di,d^2=h,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,e^b=ehi,e^c=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265355, #72265468, #72265472
maximal normal subgroup of	#72265355, #72265468, #72265472

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27503);