#116039

order	512
nilpotency class	3
isoclinism	isoclinic class of #10280
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=di,c^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,e^a=ei,f^a=fh,c^b=cf,d^c=dh,e^c=ei,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23235);