#378904

order	512
nilpotency class	3
isoclinism	isoclinic class of #28182
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

913 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=e,d^2=eh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,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, 286100);