#217827

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	#1999	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

731 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ce,d^b=di,d^c=di,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 125023);