#250007

order	512
nilpotency class	3
isoclinism	isoclinic class of #249719
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1049 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)(3,11)(4,12)(5,10)(6,9)(7,14)(8,16)(13,15)(19,25)(20,27)(22,30)(23,28)(26,29)(31,32);(1,3)(2,4)(5,7)(6,13)(8,15)(9,14)(10,11)(12,16)(17,18,21,24)(19,22,26,31)(20,28,27,23)(25,32,29,30);(1,4)(2,7)(3,8)(5,11)(6,14)(9,12)(10,13)(15,16)(17,19)(18,22)(20,29)(21,26)(23,32)(24,31)(25,27)(28,30);(1,4)(2,3)(5,15)(6,14)(7,8)(9,13)(10,12)(11,16)(20,27)(23,28)(25,29)(30,32)

Polycyclic presentation:

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

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, 157203);