#181325

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

1460 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,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, #72235686, #72239078, …
maximal normal subgroup of	#72235627, #72235686, #72239078, …
maximal quotient of	#72239123

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88521);