#110752

order	512
nilpotency class	3
isoclinism	isoclinic class of #9758
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#390
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

786 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 202 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=h,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,c^b=ch,d^b=di

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, 17948);