#181327

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	C₂³×C₄	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1466 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=hi,e^2=hi,f^2=hi,g^2=hi,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	#72235627
maximal normal subgroup of	#72235627

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88523);