#479970

order	512
nilpotency class	2
isoclinism	isoclinic class of #180869
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂⁴	#97
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

508 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=g,f^2=h,h^2=i,b^a=bh,d^a=dg,f^a=fi,c^b=cg,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287162, #72287171, #72287257, …
maximal normal subgroup of	#72287162, #72287171, #72287257, …
maximal quotient of	#72343334, #72343410, #72343804, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387166);