#108367

order	512
nilpotency class	3
isoclinism	isoclinic class of #108352
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

768 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=ei

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, 15563);