#202837

order	512
nilpotency class	3
isoclinism	isoclinic class of #19304
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

818 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)(6,10)(7,8)(11,12)(13,14,19,22)(15,21,25,27)(16,20,26,28)(17,24,18,23);(1,3,4,2)(13,15,18,26,19,25,17,16)(14,20,24,27,22,28,23,21);(5,6)(7,10)(8,12)(9,11)(13,16,17,25,19,26,18,15)(14,21,23,28,22,27,24,20);(5,7,9,8)(6,10,11,12)(14,22)(15,25)(16,26)(23,24)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110033);