#105900

order	512
nilpotency class	3
isoclinism	isoclinic class of #8453
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1152 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=di,f^b=fhi,e^c=ei,f^c=fhi

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, 13096);