#125035

order	512
nilpotency class	4
isoclinism	isoclinic class of #124996
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1543
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle	C₂	#7649
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#97 ◁ G
LECS	G ▷ C₄³ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

265 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=e,c^2=fh,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=cfgh,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32231);