#97032

order	512
nilpotency class	3
isoclinism	isoclinic class of #97030
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 ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

594 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4228);