#376658

order	512
nilpotency class	3
isoclinism	isoclinic class of #24987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2420
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14184
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

847 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=eh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,f^c=fi

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, 283854);