#6345864

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#61436	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1653 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,f^a=fh,f^b=fh,e^d=ehi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216790, #72216888, #72216897, …
maximal normal subgroup of	#72216790, #72216888, #72216897, …
maximal quotient of	#72286059, #72286341, #72343692

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253060);