#97248

order	512
nilpotency class	3
isoclinism	isoclinic class of #97214
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	#36961	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 755 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) = 56
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,f^2=hi,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, 4444);