#413465

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

988 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:

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

Polycyclic presentation:

2^9|b^2=ef,c^2=ef,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^c=ei,f^c=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, 320661);