#194285

order	512
nilpotency class	3
isoclinism	isoclinic class of #18866
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1454 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

(2,4)(5,6,8,11,9,12,7,10)(13,14,17,23,18,21,16,22)(15,20,26,19,24,27,25,28);(1,2,3,4)(6,10)(7,8)(11,12)(13,15)(14,19)(16,26)(17,25)(18,24)(20,23)(21,28)(22,27);(6,11)(7,8)(10,12)(13,15)(14,20)(16,25)(17,26)(18,24)(19,23)(21,27)(22,28);(5,7,9,8)(6,10,12,11)(14,21)(15,24)(22,23)(25,26)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dhi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101481);