#220954

order	512
nilpotency class	3
isoclinism	isoclinic class of #19883
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3283	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1142 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)(5,6)(7,10)(8,11)(9,12)(14,19)(16,17)(20,26)(21,25)(22,23)(27,28);(5,7,9,8)(6,10,12,11)(13,14)(15,20)(16,19)(17,25)(18,21)(22,26)(23,28)(24,27);(6,11)(7,8)(10,12)(13,15,18,24)(14,20,21,27)(16,22,17,23)(19,26,25,28);(6,12)(10,11)(13,16)(15,22)(17,18)(19,25)(23,24)(26,28)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fi,c^b=ci,d^b=de,e^b=eh,f^c=fi,e^d=eh

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, 128150);