#453871

order	512
nilpotency class	3
isoclinism	isoclinic class of #425083
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2576
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16181
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

545 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=efg,b^2=gh,c^2=e,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361067);