#248109

order	512
nilpotency class	3
isoclinism	isoclinic class of #248071
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1117 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:

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

Polycyclic presentation:

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

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, 155305);