#195213

order	512
nilpotency class	3
isoclinism	isoclinic class of #18192
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1440 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)(5,6,9,12,10,7,8,11)(13,14,18,23,19,24,17,22)(15,25,27,20,16,28,26,21);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,15,19,16)(14,20,24,21)(17,27,18,26)(22,28,23,25);(5,6,10,7)(8,12,9,11)(13,16,19,15)(14,21,24,20)(17,27,18,26)(22,28,23,25);(5,8,10,9)(6,11,7,12)(15,16)(20,21)(25,28)(26,27)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=i,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,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

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102409);