#97558

order	512
nilpotency class	3
isoclinism	isoclinic class of #97550
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	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

826 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 90 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)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22)(25,26,29,33,30,34,40,46)(27,31,37,43,36,42,50,54)(28,32,39,45,38,44,51,55)(35,47,49,56,48,41,52,53);(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,31)(28,35)(29,37)(30,36)(32,41)(33,43)(34,42)(38,48)(39,49)(40,50)(44,47)(45,53)(46,54)(51,52)(55,56);(1,3)(2,5)(4,8)(6,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28)(26,32)(27,35)(29,39)(30,38)(31,41)(33,45)(34,44)(36,48)(37,49)(40,51)(42,47)(43,53)(46,55)(50,52)(54,56)

Polycyclic presentation:

2^9|a^2=f,c^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,e^b=eh,f^b=fg,d^c=dh,f^c=fi

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, 4754);