#118120

order	512
nilpotency class	3
isoclinism	isoclinic class of #10637
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#397
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

710 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25316);