#195222

order	512
nilpotency class	3
isoclinism	isoclinic class of #18162
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1604 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 102418);