#93186

order	512
nilpotency class	4
isoclinism	isoclinic class of #93181
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂³×C₄	C₄²
Frattini, cosocle	C₂²×C₂²⋊Q₈	C₂²
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₂²⋊Q₈ ▷ C₂⁴ ▷ C₂² ▷ C₁

380 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 40 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 — 2 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=h,d^2=h,e^2=i,b^a=bc,c^a=cf,e^a=egh,f^a=fh,g^a=gi,c^b=cg,d^b=dfh,f^b=fi,g^b=ghi,d^c=dh,e^c=ehi,e^d=ei

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, 382);