#253308

order	512
nilpotency class	3
isoclinism	isoclinic class of #253267
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

1032 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:

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

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,d^b=di,g^b=gh,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160504);