#154069

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
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

2394 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:

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

Polycyclic presentation:

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

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, 61265);