#116404

order	512
nilpotency class	3
isoclinism	isoclinic class of #10518
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

732 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)(5,6,9,12,10,7,8,11)(13,14)(15,19)(16,18)(17,24)(20,23)(21,27)(22,26)(25,28);(14,18)(15,21,22,25)(19,27,26,28)(20,24);(2,4)(5,7,10,6)(8,11,9,12)(13,15)(14,19)(16,22)(17,25)(18,26)(20,27)(21,23)(24,28)

Polycyclic presentation:

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

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, 23600);