#482764

order	512
nilpotency class	4
isoclinism	isoclinic class of #482728
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#41932	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×#546 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1136 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)(3,8)(4,16,18,7)(5,11)(6,12)(9,17,21,31)(10,19)(13,26)(14,23)(15,22,28,20)(24,29,32,30)(25,27);(2,7)(3,13)(4,17)(8,22)(9,11)(10,21)(12,16)(14,27)(15,30)(18,31)(20,26)(23,32)(24,25)(28,29);(1,3)(2,8)(4,15)(5,14)(6,13)(7,20)(9,24)(10,25)(11,23)(12,26)(16,22)(17,29)(18,28)(19,27)(21,32)(30,31);(3,14)(8,23)(13,27)(15,29)(20,24)(22,32)(25,26)(28,30)

Polycyclic presentation:

2^9|a^2=f,e^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fgh,d^c=dh,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389960);