#209403

order	512
nilpotency class	3
isoclinism	isoclinic class of #23876
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	C₄×#120
prime radical	#537	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

840 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:

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

Polycyclic presentation:

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

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, 116599);