#132596

order	512
nilpotency class	4
isoclinism	isoclinic class of #132509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

504 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,10,12,7)(4,6)(5,9)(8,13)(11,15,16,14)(17,18,22,27,23,28,20,32)(19,24,29,39,31,40,36,47)(21,26,33,41,35,43,34,25)(30,38,45,37,46,42,44,48);(2,6)(4,13)(7,14)(8,9)(10,15)(11,16)(17,19,23,31)(18,24,28,40)(20,29,22,36)(21,30,35,46)(25,37,41,48)(26,42,43,38)(27,39,32,47)(33,44,34,45);(1,3,5,12)(2,7,9,10)(4,11,13,16)(6,14,8,15)(17,20,23,22)(18,25,28,41)(19,29,31,36)(21,34,35,33)(24,37,40,48)(26,27,43,32)(30,45,46,44)(38,47,42,39)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=eghi,f^b=fgh,d^c=dgh,e^c=ei,f^c=fgi,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39792);