#184325

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄²×D₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

695 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322026, #72322056, #72322134
maximal normal subgroup of	#72322026, #72322056, #72322134

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91521);