#115607

order	512
nilpotency class	3
isoclinism	isoclinic class of #115600
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#60801	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

846 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 134 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)(3,11,14,16)(4,7)(5,9)(6,10)(8,13,15,12)(17,18,22,26,23,28,21,27)(19,29,31,24,32,39,37,40)(20,25,35,42,36,43,34,44)(30,41,45,38,46,48,33,47);(2,7)(4,5)(8,11)(9,10)(12,13)(15,16)(17,19)(18,24)(20,33)(21,37)(22,31)(23,32)(25,41)(26,39)(27,29)(28,40)(30,35)(34,46)(36,45)(38,42)(43,48)(44,47);(1,3)(2,8)(4,13)(5,12)(6,14)(7,11)(9,16)(10,15)(17,20)(18,25)(19,30)(21,34)(22,35)(23,36)(24,38)(26,42)(27,44)(28,43)(29,41)(31,45)(32,46)(33,37)(39,48)(40,47)

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=i,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=dh,e^b=ei,f^b=fhi,d^c=di,e^c=ei,f^c=fhi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22803);