#245998

order	512
nilpotency class	3
isoclinism	isoclinic class of #245903
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	#50691	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1121 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153194);