#251123

order	512
nilpotency class	3
isoclinism	isoclinic class of #251031
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3475	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1028 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 158319);