#93189

order	512
nilpotency class	4
isoclinism	isoclinic class of #93181
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

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

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

normal subgroups only

This group has 388 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)(3,9,11,16)(4,12,14,7)(5,8)(6,10)(13,15)(17,18,20,24,22,26,32,41)(19,23,28,36,27,38,30,39)(21,33,35,40,34,25,31,42)(29,37,45,48,43,46,44,47);(1,3)(2,6,8,15)(4,13,14,10)(5,11)(7,16)(9,12)(17,19,21,29,22,27,34,43)(18,23,25,37,26,38,33,46)(20,30,31,45,32,28,35,44)(24,39,40,48,41,36,42,47)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=h,d^2=i,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, 385);