#6342646

order	512
nilpotency class	3
isoclinism	isoclinic class of #17392
automorphism	Aut(G) order 262144
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	C₂×#3444	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3208 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=hi,g^2=i,b^a=bf,d^a=di,f^a=fh,c^b=ci,g^b=gh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239232, #72239275, #72239287, …
maximal normal subgroup of	#72239232, #72239275, #72239287, …
maximal quotient of	#72239262, #72239638, #72239878

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6249842);