#6345227

order	512
nilpotency class	3
isoclinism	isoclinic class of #60750
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	#3186	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343047
maximal normal subgroup of	#72343047
maximal quotient of	#72343868, #72343888

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252423);