#97094

order	512
nilpotency class	3
isoclinism	isoclinic class of #97080
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, prime radical	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

898 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 898 conjugacy classes of subgroups — too many to draw. Only its 114 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18)(19,22)(20,27,28,23)(21,24)(25,31,32,29)(26,30)(33,34,36,40,37,39,45,47)(35,41,43,48,42,38,44,46);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31)(33,35,37,42)(34,38,39,41)(36,44,45,43)(40,48,47,46);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31)(34,39)(35,42)(36,45)(46,48)

Polycyclic presentation:

2^9|a^2=f,b^2=ghi,c^2=gi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=ci,f^b=fg,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4290);