#210653

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
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 ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

724 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)(5,6,8,11)(7,9,12,10)(13,14,17,21,18,16,20,25)(15,19,23,26,22,24,27,28);(1,2,3,4)(13,15)(14,19)(16,24)(17,23)(18,22)(20,27)(21,26)(25,28);(6,9)(7,12)(10,11)(15,22)(19,24)(23,27)(26,28);(13,16,17,25,18,14,20,21)(15,19,23,26,22,24,27,28)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=gh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^b=di,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, 117849);