#127727

order	512
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×#102
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 52 points:

(1,2,3,4)(5,6)(7,14)(8,12)(9,17)(10,13)(11,15)(16,19)(18,20)(21,22)(23,32)(24,37)(25,40)(26,42)(27,31)(28,33)(29,39)(30,38)(34,44)(35,52)(36,48)(41,45)(43,51)(46,50)(47,49);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,20)(14,17)(21,23,25,34,38,49,39,50,27,36,24,33,26,35,41,51)(22,28,29,32,42,46,40,52,31,44,45,48,30,43,37,47);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=gi,f^2=ghi,g^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,d^b=dh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34923);