#202740

order	512
nilpotency class	3
isoclinism	isoclinic class of #19230
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

826 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=e,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,f^c=fhi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109936);