#192269

order	512
nilpotency class	3
isoclinism	isoclinic class of #18192
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

891 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:

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

Polycyclic presentation:

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

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, 99465);