#116176

order	512
nilpotency class	3
isoclinism	isoclinic class of #116174
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

585 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,f^2=hi,g^2=h,b^a=bd,c^a=ce,e^a=ei,f^a=fhi,c^b=cf,e^b=ei,f^b=fi,d^c=dh,f^c=fhi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23372);