#248139

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	#24176	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1160 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)(3,11)(4,7)(5,15)(6,10)(8,13)(9,12)(14,16)(17,18)(19,24)(20,29)(21,25)(22,30)(23,26)(27,31)(28,32);(1,3,6,13)(2,7,10,16)(4,5,14,12)(8,15,11,9)(17,19,23,28)(18,24,26,32)(20,21,27,22)(25,31,30,29);(17,20)(18,24)(19,21)(22,28)(23,27)(25,31)(26,32)(29,30);(1,4,6,14)(2,8,10,11)(3,12,13,5)(7,9,16,15)(17,21,23,22)(18,25,26,30)(19,27,28,20)(24,31,32,29)

Polycyclic presentation:

2^9|b^2=hi,d^2=hi,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, 155335);