#116344

order	512
nilpotency class	3
isoclinism	isoclinic class of #9938
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

691 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239073, #72249689, #72249691
maximal normal subgroup of	#72239073, #72249689, #72249691
maximal quotient of	#72257605

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23540);