#192438

order	512
nilpotency class	3
isoclinism	isoclinic class of #18376
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 20 points:

(1,2,4,3)(6,11)(7,12)(8,9)(14,19)(15,20)(16,17);(1,3,4,2)(5,6,10,7)(8,12,9,11)(13,14,18,15)(16,19,17,20);(5,7,10,6)(8,12,9,11)(13,15,18,14)(16,20,17,19);(6,7)(11,12)(13,16,18,17)(14,20,15,19)

Polycyclic presentation:

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

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, 99634);