#93118

order	512
nilpotency class	4
isoclinism	isoclinic class of #93109
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂³×C₄	C₄²
Frattini, cosocle	C₂²×#125	C₂²
prime radical	C₂⁴	#95
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#125 ◁ G
LECS	G ▷ C₂²×#125 ▷ C₂⁴ ▷ C₂² ▷ C₁

299 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 64 points:

(1,2)(3,11,13,22)(4,15,17,8)(5,9)(6,10)(7,12)(14,26,29,32)(16,21)(18,25,27,23)(19,24)(20,28)(30,31)(33,34,37,41,38,42,51,59)(35,43,45,60,46,55,63,64)(36,47,48,58,49,40,50,56)(39,52,53,44,54,61,57,62);(1,3,5,14,6,13,19,29)(2,7,9,20,10,21,24,31)(4,16,18,30,17,12,27,28)(8,22,23,32,15,11,25,26)(33,35)(34,39)(36,44)(37,45,51,63)(38,46)(40,55)(41,57,59,53)(42,54)(43,47)(48,61,50,52)(49,62)(56,60,58,64)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=h,d^2=i,e^2=hi,b^a=bc,c^a=cf,e^a=egh,f^a=fh,g^a=gi,c^b=cg,d^b=dfh,f^b=fi,d^c=dh,e^d=ei

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 314);