#188809

order	512
nilpotency class	3
isoclinism	isoclinic class of #8608
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1056 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18)(19,25,27,22)(20,23)(21,24)(26,31,32,29)(28,30);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16);(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285898, #72285955, #72285976, …
maximal normal subgroup of	#72285898, #72285955, #72285976, …
maximal quotient of	#72287291, #72287362

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 96005);