#189130

order	512
nilpotency class	3
isoclinism	isoclinic class of #19368
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

976 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 96326);