#208601

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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

857 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)(9,10,14,15,13,16,12,11)(18,22)(19,25)(20,27)(23,30)(24,29)(28,31);(9,11,14,16,13,15,12,10)(17,18,20,22,21,24,27,29)(19,23,26,28,25,31,32,30);(2,5)(6,8)(9,12,13,14)(10,15,16,11)(17,19)(18,23)(20,26)(21,25)(22,28)(24,31)(27,32)(29,30);(1,3,4,7)(2,6,5,8)(19,25)(23,31)(26,32)(28,30)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,d^2=fh,e^2=hi,g^2=i,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, 115797);