#148284

order	512
nilpotency class	4
isoclinism	isoclinic class of #13309
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2034
prime radical	C₂⁵	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

732 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249927, #72249939, #72249947, …
maximal normal subgroup of	#72249927, #72249939, #72249947, …
maximal quotient of	#72265380

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55480);