#110003

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#386
prime radical	C₂⁴	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

500 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,d^b=dh,e^b=ei,d^c=di,e^c=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17199);