#504432

order	512
nilpotency class	4
isoclinism	isoclinic class of #504384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13341	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1771 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

593 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)(5,6,10,12)(7,16,19,13)(8,20,23,14)(9,25,15,11)(17,32,34,28)(18,26,30,33)(21,29,36,35)(22,24,27,31);(2,4)(5,7)(6,13)(8,21)(9,18)(10,19)(11,26)(12,16)(14,29)(15,30)(17,24)(20,35)(22,32)(23,36)(25,33)(27,28)(31,34);(5,8,10,23)(6,14,12,20)(7,17,19,34)(9,24,15,31)(11,27,25,22)(13,28,16,32)(18,35,30,29)(21,33,36,26);(2,4)(5,9,6,11,10,15,12,25)(7,18,13,26,19,30,16,33)(8,22,20,31,23,27,14,24)(17,21,32,35,34,36,28,29)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321245
maximal normal subgroup of	#72321245

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411628);