#210494

order	512
nilpotency class	3
isoclinism	isoclinic class of #32884
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	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

706 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)(5,6,8,7,9,10,12,11)(13,14,18,23)(15,24,16,20)(17,22,19,27)(21,25,28,26);(1,2,3,4)(13,15,18,16)(14,20,23,24)(17,26,19,25)(21,22,28,27);(13,16,18,15)(14,21,23,28)(17,26,19,25)(20,22,24,27);(5,7,8,10,9,11,12,6)(13,17,18,19)(14,22,23,27)(15,25,16,26)(20,21,24,28)

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,d^b=dh,f^c=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 117690);