#377767

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1113 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)(7,17)(8,22,24,28)(9,14)(10,15)(11,16)(12,20)(13,23,29,25)(18,26,32,36)(19,30)(21,27)(31,33,35,34);(1,2)(3,4)(5,7,9,20,11,19,10,21)(6,12,15,17,16,27,14,30)(8,18,23,33,24,32,25,34)(13,26,28,31,29,36,22,35);(2,4)(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,22)(15,28)(16,29)(17,31)(19,32)(20,33)(21,34)(27,36)(30,35);(6,14,16,15)(7,19)(12,17,27,30)(13,22,29,28)(18,32)(20,21)(26,31,36,35)(33,34)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=efh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250732, #72252980, #72256873
maximal normal subgroup of	#72250732, #72252980, #72256873

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284963);