#492927

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

710 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,10,15,9,17,25,35,11,16,27,34,26,14,19,28)(7,18,22,33,21,12,24,30,20,31,36,13,32,29,8,23);(1,2,3,4)(5,7)(6,12,16,29)(8,19)(9,21)(10,22)(11,20)(13,28,33,35)(14,18,17,31)(15,30,34,23)(24,25)(26,32)(27,36);(5,8,9,22,11,24,26,36)(6,13,14,30,16,33,17,23)(7,19,21,10,20,25,32,27)(12,28,18,34,29,35,31,15);(7,20)(8,24)(12,29)(13,33)(18,31)(21,32)(22,36)(23,30)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400123);