#230387

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2361
prime radical	#19128	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

540 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322005, #72322056, #72322213
maximal normal subgroup of	#72322005, #72322056, #72322213

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137583);