#6472890

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470874
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1733 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6380086);