#192829

order	512
nilpotency class	3
isoclinism	isoclinic class of #18866
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1008 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)(6,11)(7,12)(8,9)(13,14,18,22)(15,23,16,20)(17,26,27,21)(19,25,28,24);(1,3,4,2)(5,6,10,7)(8,11,9,12)(13,15,18,16)(14,19,22,28)(17,25,27,24)(20,26,23,21);(5,7,10,6)(8,12,9,11)(13,16,18,15)(14,20,22,23)(17,25,27,24)(19,26,28,21);(5,8,10,9)(6,12,7,11)(15,16)(19,28)(20,23)(24,25)

Polycyclic presentation:

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

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, 100025);