#394685

order	512
nilpotency class	3
isoclinism	isoclinic class of #394685
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14718
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3019	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2185 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)(3,13)(4,17)(5,11)(6,10)(7,12)(8,15)(9,20)(14,26)(16,24)(18,31)(19,27)(21,28)(22,25)(23,32)(29,30);(1,3)(2,8)(4,18)(5,22)(6,15)(7,16)(9,26)(10,24)(11,13)(12,25)(14,19)(17,29)(20,32)(21,30)(23,28)(27,31);(1,4,5,19,7,20,6,21)(2,9,10,27,12,17,11,28)(3,14,15,18,16,30,22,32)(8,23,13,29,25,31,24,26);(1,3)(2,8)(4,14)(5,15)(6,22)(7,16)(9,23)(10,13)(11,24)(12,25)(17,31)(18,19)(20,30)(21,32)(26,28)(27,29)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216071, #72216142, #72216154, …
maximal normal subgroup of	#72216071, #72216142, #72216154, …
maximal quotient of	C₂×#394685, #72252094, #72265357

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301881);