#93062

order	512
nilpotency class	4
isoclinism	isoclinic class of #93047
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1294
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂×#431	C₂²
prime radical	C₂⁴	#107
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×#431 ▷ C₂²×C₄ ▷ C₂² ▷ C₁

222 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 58 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 — 2 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=hi,e^2=hi,f^2=h,g^2=h,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=gh,c^b=cg,d^b=dh,f^b=fh,g^b=gi,e^c=ehi

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, 258);