#210437

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

746 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=h,d^2=gh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^c=dh,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, 117633);