#202703

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=e,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,c^b=ch,d^b=dhi,d^c=dhi

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, 109899);