#253303

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×#1595	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1103 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,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, 160499);