#230071

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2361
prime radical	C₂×AGL(1,ℤ/8)	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

567 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=eh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ci,d^b=de,e^b=eh,g^b=gh,d^c=di,f^c=fh

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, 137267);