#110421

order	512
nilpotency class	3
isoclinism	isoclinic class of #9781
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#390
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1086 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17617);