#132660

order	512
nilpotency class	4
isoclinism	isoclinic class of #132573
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) = 128

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

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=ci,d^b=dh,e^b=egh,f^b=fgh,d^c=dghi,e^c=ei,f^c=fg,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, 39856);