#360582

order	512
nilpotency class	3
isoclinism	isoclinic class of #24408
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13877
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2149 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319834, #72338478, #72338484, …
maximal normal subgroup of	#72319834, #72338478, #72338484, …
maximal quotient of	#72343070

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267778);