#108666

order	512
nilpotency class	3
isoclinism	isoclinic class of #8760
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#381
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

796 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,e^2=hi,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^b=fh,e^c=ei,f^c=fh

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, 15862);