#6345826

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
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	#61439	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2922 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1678 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260382, #72321275, #72343921
maximal normal subgroup of	#72260382, #72321275, #72343921
maximal quotient of	#72343943, #72343945

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253022);