#192919

order	512
nilpotency class	3
isoclinism	isoclinic class of #18712
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

870 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:

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

Polycyclic presentation:

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

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, 100115);