#493252

order	512
nilpotency class	4
isoclinism	isoclinic class of #33469
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

707 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gi,d^2=f,e^2=hi,h^2=i,b^a=be,c^a=ch,e^a=eh,g^a=gi,h^a=hi,e^b=ehi,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#33469, #72320189, #72320393, …
maximal normal subgroup of	C₄×#33469, #72320189, #72320393, …
maximal quotient of	#72343996

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400448);