#380136

order	512
nilpotency class	3
isoclinism	isoclinic class of #29902
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2622
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14259
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1144 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338478, #72338488, #72338495
maximal normal subgroup of	#72338478, #72338488, #72338495

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287332);