#150649

order	512
nilpotency class	5
isoclinism	isoclinic class of #150594
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#390	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ #2309 ◁ G
LECS	G ▷ #390 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

339 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 52 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18)(19,28,32,39)(20,33)(21,26,29,25)(22,27)(23,31,38,43)(24,35)(30,45,46,37)(34,42,48,47)(36,41,44,40);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19,29,43,22,31,21,32)(18,23,25,38,27,39,26,28)(20,34,44,46,35,30,36,48)(24,37,41,45,33,47,40,42);(2,7)(6,14)(8,15)(9,16)(17,20)(18,24)(19,30)(21,36)(22,35)(23,37)(25,41)(26,40)(27,33)(28,42)(29,44)(31,34)(32,46)(38,45)(39,47)(43,48)

Polycyclic presentation:

2^9|a^2=e,b^2=fi,d^2=gh,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ch,d^b=dg,e^b=efghi,d^c=di,e^d=ehi,f^e=fi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57845);