#369550

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

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	#2420
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

736 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)(7,18)(8,22)(9,13)(10,27)(11,16)(12,17)(14,25)(15,21)(19,28)(20,29)(23,30)(24,32)(26,31)(33,34)(35,36);(1,3,4,2)(5,7,10,9,12,21,11,20)(6,13,16,18,17,29,27,15)(8,23,25,19,26,34,24,35)(14,30,22,36,32,33,31,28);(5,8,10,24,12,26,11,25)(6,14,27,22,17,32,16,31)(7,19,9,23,21,35,20,34)(13,28,15,30,29,36,18,33);(5,9,10,21,12,20,11,7)(6,15,16,13,17,18,27,29)(8,23,24,35,26,34,25,19)(14,30,31,28,32,33,22,36)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=efg,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,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, 276746);