#465624

order	512
nilpotency class	3
isoclinism	isoclinic class of #455564
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1093 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)(7,11,18,26)(8,19,20,13)(9,12,22,14)(16,24,29,32)(17,23,31,25)(21,28,33,27)(30,34,36,35);(1,2,3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,23)(13,24)(14,25)(15,26)(19,32)(20,29)(21,30)(22,31)(27,34)(28,35)(33,36);(2,4)(6,12)(8,20)(9,22)(11,23)(13,28)(14,15)(16,29)(17,31)(19,27)(24,35)(25,26)(32,34);(5,8,9,21,10,20,22,33)(6,13,14,28,15,19,12,27)(7,16,17,30,18,29,31,36)(11,24,25,35,26,32,23,34)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,h^b=hi,d^c=de,f^c=fi,e^d=ei,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, 372820);