#289223

order	512
nilpotency class	3
isoclinism	isoclinic class of #289219
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
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

569 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)(3,11)(4,7)(5,14)(6,10)(8,12)(9,15)(13,16)(17,18,22,24,21,26,20,25)(19,23,29,32,27,31,28,30);(1,3,5,4,6,12,15,13)(2,7,9,11,10,16,14,8)(17,19,20,28,21,27,22,29)(18,23,25,30,26,31,24,32);(18,24)(19,27)(20,22)(23,30)(25,26)(31,32);(1,4,5,12,6,13,15,3)(2,8,9,7,10,11,14,16)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196419);