#206464

order	512
nilpotency class	3
isoclinism	isoclinic class of #19671
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	#60782	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1012 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285558, #72322077, #72322149
maximal normal subgroup of	#72285558, #72322077, #72322149

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113660);