#108214

order	512
nilpotency class	3
isoclinism	isoclinic class of #9467
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287575, #72343130, #72343141
maximal normal subgroup of	#72287575, #72343130, #72343141
maximal quotient of	#72287815, #72287825

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15410);