#249769

order	512
nilpotency class	3
isoclinism	isoclinic class of #249759
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

1100 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,27)(20,28)(21,24)(22,25)(23,26)(29,31)(30,32);(4,5)(8,13)(10,15)(11,14)(17,19,23,30)(18,20,26,31)(21,29,22,28)(24,27,25,32);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(18,24)(19,28)(21,22)(25,26)(27,32)(29,30);(1,4)(2,8)(3,11)(5,6)(7,15)(9,13)(10,16)(12,14)(17,20)(18,19)(21,32)(22,27)(23,31)(24,28)(25,29)(26,30)

Polycyclic presentation:

2^9|b^2=h,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,d^b=di,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156965);