#107998

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#386
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=di,e^b=eh,f^b=fhi,d^c=dh,e^c=ei,f^c=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15194);