#125034

order	512
nilpotency class	4
isoclinism	isoclinic class of #124850
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
socle	C₂²	#1466
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

278 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 68 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18,20,25,21,26,32,40,22,23,31,35,33,36,44,47)(19,27,29,41,30,24,37,38,28,39,42,48,43,34,45,46);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(18,23)(19,28)(26,36)(29,42)(30,43)(37,45)(38,46)(41,48);(2,7)(6,14)(8,15)(9,16)(17,19,21,30,22,28,33,43)(18,24,36,27,23,34,26,39)(20,29,32,37,31,42,44,45)(25,38,47,41,35,46,40,48)

Polycyclic presentation:

2^9|a^2=e,c^2=fh,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=cfgh,d^a=df,f^a=fi,c^b=ci,d^b=dg,e^b=efg,e^d=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, 32230);