#253260

order	512
nilpotency class	3
isoclinism	isoclinic class of #253219
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#1600	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1056 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,g^b=gh,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160456);