#230436

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	#19157	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

604 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,7,3,6,5,4,8)(9,10)(11,21,27,37)(12,13)(14,18)(15,20)(16,25,36,26)(17,29)(19,22)(23,34,40,24)(28,32)(30,33)(31,39,38,35);(1,3,6,8)(2,4,5,7)(9,11,22,25,15,27,14,26)(10,16,19,21,20,36,18,37)(12,23,29,39,30,40,28,35)(13,31,17,34,33,38,32,24);(1,4,6,7)(2,3,5,8)(9,12,15,30)(10,17,20,32)(11,23,27,40)(13,19,33,18)(14,28,22,29)(16,34,36,24)(21,38,37,31)(25,39,26,35);(1,5,4,3,6,2,7,8)(9,13,14,32,15,33,22,17)(10,12,18,28,20,30,19,29)(11,24,25,31,27,34,26,38)(16,35,21,23,36,39,37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=e,e^2=h,f^2=i,g^2=hi,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, 137632);