#153991

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
commutator	C₂³	C₂²×C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,4,3)(5,6,8,7)(18,21)(19,23)(25,26,28,29)(27,30,31,32);(5,6,8,7)(10,13)(14,16)(18,21)(22,24)(26,29)(27,31);(1,3,4,2)(5,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,30)(28,29)(31,32);(9,11,12,15)(10,14,13,16)(17,19)(18,22)(20,23)(21,24)(25,27,28,31)(26,30,29,32)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=eh,d^2=efg,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de

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, 61187);