#379800

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2348 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313677, #72313722
maximal normal subgroup of	#72313677, #72313722
maximal quotient of	#72314156, #72314205

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286996);