#6345256

order	512
nilpotency class	3
isoclinism	isoclinic class of #60768
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	#60944	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1584 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:

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

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,d^a=dh,e^a=ehi,f^a=fh,d^b=di,f^b=fh,d^c=dh,e^c=eh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318537, #72319587, #72319674
maximal normal subgroup of	#72318537, #72319587, #72319674

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252452);