#183554

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

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

860 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)(5,6,9,7,8,11,10,12)(13,14,17,16,18,20,25,24)(15,19,23,21,22,26,28,27);(2,4)(5,7,9,11,8,12,10,6)(13,15)(14,19)(16,21)(17,23)(18,22)(20,26)(24,27)(25,28);(5,6,10,12,8,11,9,7)(13,16,17,20,18,24,25,14)(15,21,23,26,22,27,28,19);(6,11)(7,12)(15,22)(19,26)(21,27)(23,28)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=di,d^b=dh,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90750);