#111206

order	512
nilpotency class	3
isoclinism	isoclinic class of #9239
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efi,d^2=hi,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18402);