#384927

order	512
nilpotency class	3
isoclinism	isoclinic class of #31076
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2684
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

844 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=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, 292123);