#409168

order	512
nilpotency class	3
isoclinism	isoclinic class of #394469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

775 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316364);