#97556

order	512
nilpotency class	3
isoclinism	isoclinic class of #97550
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂²×H(5,𝔽₂)	C₄
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

1566 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 1566 conjugacy classes of subgroups — too many to draw. Only its 98 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)(3,7,8,5)(4,6)(9,10)(11,15,16,13)(12,14)(17,18,21,25,22,26,32,38)(19,23,29,35,28,34,42,46)(20,24,31,37,30,36,43,47)(27,39,41,48,40,33,44,45);(1,3)(2,5)(4,8)(6,7)(17,19)(18,23)(20,27)(21,29)(22,28)(24,33)(25,35)(26,34)(30,40)(31,41)(32,42)(36,39)(37,45)(38,46)(43,44)(47,48);(9,11)(10,13)(12,16)(14,15)(17,20)(18,24)(19,27)(21,31)(22,30)(23,33)(25,37)(26,36)(28,40)(29,41)(32,43)(34,39)(35,45)(38,47)(42,44)(46,48)

Polycyclic presentation:

2^9|a^2=f,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.

Relationships

maximal subgroup of	#72227712, #72227722, #72227724
maximal normal subgroup of	#72227712, #72227722, #72227724

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4752);