#379581

order	512
nilpotency class	3
isoclinism	isoclinic class of #24986
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

930 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286777);