#251727

order	512
nilpotency class	3
isoclinism	isoclinic class of #251701
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2619	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

852 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=ehi,c^2=i,d^2=i,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,d^c=di,e^c=eh,f^c=fi,g^c=gh,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, 158923);