#181151

order	512
nilpotency class	3
isoclinism	isoclinic class of #7943
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#95
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1554 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=i,e^2=hi,f^2=h,b^a=be,e^a=eh,g^a=ghi,c^b=ch,f^b=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235585, #72238767, #72238770, …
maximal normal subgroup of	#72235585, #72238767, #72238770, …
maximal quotient of	#72239977, #72257622, #72257751

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88347);