#213262

order	512
nilpotency class	3
isoclinism	isoclinic class of #19713
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#127
prime radical	#19603	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

714 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=fi,d^2=g,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,d^c=dh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120458);