#108408

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#117
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#383
prime radical	C₂⁴	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

690 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)(5,6,7,9,8,10,11,12)(13,14)(15,23,24,28)(16,26,25,20)(17,21)(18,27)(19,22);(13,15,18,25,19,24,17,16)(14,20,21,23,22,26,27,28);(2,4)(13,16,17,24,19,25,18,15)(14,20,21,23,22,26,27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15604);