#97219

order	512
nilpotency class	3
isoclinism	isoclinic class of #97214
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

694 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=g,d^2=h,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, 4415);