#368464

order	512
nilpotency class	3
isoclinism	isoclinic class of #24404
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1425 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320778, #72342923, #72342942, …
maximal normal subgroup of	#72320778, #72342923, #72342942, …
maximal quotient of	#72343052

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275660);