#132780

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	C₂³	#379
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

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

normal subgroups only

This group has 465 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=hi,d^2=hi,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=eghi,f^b=fghi,d^c=dgh,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, 39976);