#194597

order	512
nilpotency class	3
isoclinism	isoclinic class of #19050
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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×#3150	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1510 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)(5,6,10,11,9,7,8,12)(13,14,17,22,18,23,16,21)(15,24,25,20,26,28,27,19);(1,2,3,4)(5,7,9,6)(8,12,10,11)(13,15)(14,19)(16,25)(17,27)(18,26)(20,23)(21,24)(22,28);(8,10)(11,12)(13,15)(14,20)(16,27)(17,25)(18,26)(19,23)(21,24)(22,28);(5,8,9,10)(6,11,7,12)(13,16,18,17)(14,21,23,22)(15,25,26,27)(19,24,20,28)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,f^a=fh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fhi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101793);