#208580

order	512
nilpotency class	3
isoclinism	isoclinic class of #163983
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2619	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

916 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)(3,6)(4,5)(7,8)(9,10,12,15,13,14,16,11)(17,18,21,25,22,26,20,23)(19,24,29,32,27,31,28,30);(9,11,12,14,13,15,16,10)(18,23,26,25)(24,30,31,32);(2,5)(6,8)(17,19)(18,24)(20,28)(21,29)(22,27)(23,30)(25,32)(26,31);(1,3,4,7)(2,6,5,8)(19,27)(24,31)(28,29)(30,32)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=f,e^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,e^b=ei,g^b=gi,d^c=dhi

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, 115776);