#113054

order	512
nilpotency class	3
isoclinism	isoclinic class of #97690
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#35329	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1216 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1216 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,e^b=ei,g^b=ghi,d^c=di,g^c=gh

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, 20250);