#193770

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×#386
prime radical	#60700	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1135 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,e^b=eh,g^b=gh,d^c=di,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100966);