#111235

order	512
nilpotency class	3
isoclinism	isoclinic class of #96990
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

844 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 844 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18431);