#251719

order	512
nilpotency class	3
isoclinism	isoclinic class of #251701
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	C₂×C₄²	C₂⁴
prime radical	#18304	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

848 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:

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

Polycyclic presentation:

2^9|b^2=ei,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, 158915);