#461830

order	512
nilpotency class	3
isoclinism	isoclinic class of #455962
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

746 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)(3,4)(5,6,10,15,11,9,26,14)(7,16,19,28,18,12,20,29)(8,21,24,13,25,31,36,23)(17,33,35,32,34,30,22,27);(5,7,10,19,11,18,26,20)(6,12,14,28,9,16,15,29)(8,22,24,17,25,35,36,34)(13,30,21,32,23,33,31,27);(5,8,10,24,11,25,26,36)(6,13,14,21,9,23,15,31)(7,17,20,22,18,34,19,35)(12,27,29,33,16,32,28,30);(2,4)(5,9)(6,11)(7,16)(8,23)(10,15)(12,18)(13,25)(14,26)(17,27)(19,29)(20,28)(21,36)(22,33)(24,31)(30,35)(32,34)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 369026);