#363499

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1274 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=eg,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^b=fi,e^c=ei,g^c=gi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239887, #72239904, #72239910, …
maximal normal subgroup of	#72239887, #72239904, #72239910, …
maximal quotient of	#72239889

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270695);