#361295

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1738 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=i,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,e^d=ei

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, 268491);