#97892

order	512
nilpotency class	3
isoclinism	isoclinic class of #97164
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	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 505 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) = 256

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

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

Polycyclic presentation:

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

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, 5088);