#230405

order	512
nilpotency class	3
isoclinism	isoclinic class of #20158
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	#19160	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

534 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,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,d^c=dh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137601);