#388118

order	512
nilpotency class	3
isoclinism	isoclinic class of #31076
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×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	#2782
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

765 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)(5,6,12,18)(7,9,15,29)(8,22,27,33)(10,17,11,16)(13,21,28,20)(14,26,32,25)(19,34,31,30)(23,36,35,24);(5,7,11,21,12,15,10,20)(6,13,17,29,18,28,16,9)(8,23,25,19,27,35,26,31)(14,24,22,34,32,36,33,30);(5,8,10,25,12,27,11,26)(6,14,17,33,18,32,16,22)(7,19,20,35,15,31,21,23)(9,24,13,30,29,36,28,34);(5,9,10,13,12,29,11,28)(6,15,17,21,18,7,16,20)(8,24,25,30,27,36,26,34)(14,31,33,23,32,19,22,35)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=efg,d^2=efg,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=dei,f^c=fi,g^c=gi,e^d=ei,f^d=fi

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, 295314);