#355556

order	512
nilpotency class	3
isoclinism	isoclinic class of #22284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1414 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,h^a=hi,c^b=ce,f^b=fi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226670, #72226671, #72226672
maximal normal subgroup of	#72226670, #72226671, #72226672

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262752);