#97239

order	512
nilpotency class	3
isoclinism	isoclinic class of #97214
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

646 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 56 points:

(1,2)(3,7,8,5)(4,6)(9,10,15,17,14,11,13,19)(12,16,22,23,21,18,20,24)(25,26)(27,35,38,47)(28,39,41,32)(29,33)(30,34)(31,37,45,42)(36,50)(40,54,56,46)(43,48)(44,51)(49,52)(53,55);(1,3,4,8)(2,5,6,7)(9,11,14,10)(12,18,21,16)(13,17,15,19)(20,23,22,24)(25,27)(26,31)(28,36)(29,42)(30,38)(32,44)(33,47)(34,45)(35,48)(37,43)(39,53)(40,55)(41,52)(46,50)(49,54)(51,56);(9,12,14,21)(10,16,11,18)(13,22,15,20)(17,24,19,23)(25,28,30,41)(26,32,34,39)(27,36,38,52)(29,40,43,56)(31,44,45,53)(33,46,48,54)(35,49,47,50)(37,51,42,55)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=ei,f^b=fgh,d^c=di,f^c=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4435);