#210812

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

752 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=gh,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ch,d^b=dh,d^c=dh

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, 118008);