#364155

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
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

1802 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264907, #72264917, #72264952
maximal normal subgroup of	#72264907, #72264917, #72264952

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271351);