#97308

order	512
nilpotency class	3
isoclinism	isoclinic class of #97300
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	#2679	C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

543 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 543 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4504);