#6343125

order	512
nilpotency class	3
isoclinism	isoclinic class of #61591
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

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	C₂³×C₂²⋊C₄
prime radical	C₂×#3483	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3180 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)(3,12)(4,9)(5,10)(6,15)(7,11)(8,13)(14,16)(17,18,21,26,22,25,20,24)(19,27,29,23,30,32,28,31);(1,3,7,4)(2,8,11,16)(5,13,6,14)(9,15,12,10)(17,19,22,30)(18,23,25,31)(20,28,21,29)(24,27,26,32);(1,4,7,3)(2,9,11,12)(5,13,6,14)(8,15,16,10)(17,20,22,21)(18,24,25,26)(19,28,30,29)(23,27,31,32);(1,5,7,6)(2,10,11,15)(3,13,4,14)(8,9,16,12)(17,20,22,21)(18,24,25,26)(19,28,30,29)(23,27,31,32);(18,25)(23,31)(24,26)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303322, #72303328, #72303419, …
maximal normal subgroup of	#72303322, #72303328, #72303419, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250321);