#249755

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
prime radical	#18568	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1022 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=hi,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, 156951);