#492959

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1286 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6)(7,17,19,28)(8,14)(9,23)(10,16)(11,13)(12,18,26,21)(15,25)(20,34,33,31)(22,30)(24,29)(27,35,36,32);(2,4)(5,7,9,20,10,21,24,35,11,19,25,33,8,18,22,32)(6,12,15,27,16,28,30,34,13,26,23,36,14,17,29,31);(5,6)(7,17,19,28)(8,14)(9,23)(10,16)(11,13)(12,18,26,21)(15,25)(20,34,33,31)(22,30)(24,29)(27,35,36,32);(6,13)(12,26)(14,16)(15,23)(17,28)(27,36)(29,30)(31,34)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=i,g^2=h,h^2=i,b^a=be,d^a=di,g^a=gi,c^b=ch,f^b=fi,d^c=di,g^c=gi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237211, #72239200, #72239337, …
maximal normal subgroup of	#72237211, #72239200, #72239337, …
maximal quotient of	#72309885, #72309924

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400155);