#461304

order	512
nilpotency class	3
isoclinism	isoclinic class of #454986
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

905 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)(5,6,11,14,9,15,10,16)(7,12,25,29,18,28,19,27)(8,20,24,31,22,13,23,32)(17,33,35,30,34,26,21,36);(1,2,4,3)(5,7,9,18)(6,12,15,28)(8,21,22,35)(10,25,11,19)(13,30,20,36)(14,27,16,29)(17,24,34,23)(26,32,33,31);(2,3)(5,8)(6,13)(7,17)(9,22)(10,24)(11,23)(12,26)(14,31)(15,20)(16,32)(18,34)(19,35)(21,25)(27,36)(28,33)(29,30);(1,3,4,2)(6,14,15,16)(8,22)(12,27,28,29)(13,31,20,32)(17,34)(21,35)(23,24)(26,36,33,30)

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,d^c=de,g^c=gi,h^c=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, 368500);