#399171

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1361 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257091, #72257207, #72257218
maximal normal subgroup of	#72257091, #72257207, #72257218

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 306367);