#210626

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂×#127	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

764 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=h,d^2=g,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ci,d^c=di,f^c=fh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 117822);