#127587

order	512
nilpotency class	4
isoclinism	isoclinic class of #11836
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1514
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8018
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

372 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gh,d^2=gi,e^2=i,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,d^c=di,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286539, #72286542, #72286560
maximal normal subgroup of	#72286539, #72286542, #72286560

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34783);