#192671

order	512
nilpotency class	3
isoclinism	isoclinic class of #18618
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂³×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1180 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99867);